Gradian

28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance. Template:Infobox Polygon In Euclidean geometry, a convex quadrilateral with at least one pair of parallel sides is referred to as a trapezoid in American and Canadian English but as a trapezium in English outside North America. The parallel sides are called the bases of the trapezoid and the other two sides are called the legs or the lateral sides (if they are not parallel; otherwise there are two pairs of bases). A scalene trapezoid is a trapezoid with no sides of equal measure, in contrast to the special cases below.

There is some disagreement whether parallelograms, which have two pairs of parallel sides, should be counted as trapezoids. Some define a trapezoid as a quadrilateral having only one pair of parallel sides (the exclusive definition), thereby excluding parallelograms. Others define a trapezoid as a quadrilateral with at least one pair of parallel sides (the inclusive definition), making the parallelogram a special type of trapezoid. The latter definition is consistent with its uses in higher mathematics such as calculus. The former definition would make such concepts as the trapezoidal approximation to a definite integral ill-defined. This article uses the inclusive definition and considers parallelograms as special cases of a trapezoid. This is also advocated in the taxonomy of quadrilaterals.

Etymology

The term trapezium has been in use in English since 1570, from Late Latin trapezium, from Greek τραπέζιον (trapézion), literally "a little table", a diminutive of τράπεζα (trápeza), "a table", itself from τετράς (tetrás), "four" + πέζα (péza), "a foot, an edge". The first recorded use of the Greek word translated trapezoid (τραπέζοειδη, trapézoeide, "table-like") was by Marinus Proclus (412 to 485 AD) in his Commentary on the first book of Euclid's Elements.

This article uses the term trapezoid in the sense that is current in the United States and Canada. In all other languages using a word derived from the Greek for this figure, the form closest to trapezium (e.g. French trapèze, Italian trapezio, Spanish trapecio, German Trapez, Russian трапеция) is used.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

Special cases

In an isosceles trapezoid, the legs (AD and BC in the figure above) have the same length, and the base angles have the same measure. In a right trapezoid (also called right-angled trapezoid), two adjacent angles are right angles. A tangential trapezoid is a trapezoid that has an incircle.

A trapezoid with two pairs of parallel sides is a parallelogram. Under the inclusive definition, all parallelograms (including rhombuses, rectangles and squares) are trapezoids.

Characterizations

Given a convex quadrilateral, the following properties are equivalent, and each implies that the quadrilateral is a trapezoid:

• It has two adjacent angles that are supplementary, that is, they add up 180 degrees.

• The angle between a side and a diagonal is equal to the angle between the opposite side and the same diagonal.

• The diagonals cut each other in mutually the same ratio (this ratio is the same as that between the lengths of the parallel sides).

• The diagonals cut the quadrilateral into four triangles of which one opposite pair are similar.

• The diagonals cut the quadrilateral into four triangles of which one opposite pair have equal areas.Primarily based on the most recent URA personal property value index (PPPI) flash estimates, we know that the PPPI, which represents the overall real property price development, has dipped in 2013Q4. That is the first dip the market has seen within the final 2 years.
  
  To give you some perspective, the entire number of personal properties in Singapore (together with govt condominiums) is 297,689 in 2013Q3. Primarily based on the projection that there will be 19,302 units accomplished in 2014, the rise in residential models works out to be more than 6%. With a lot New Ec Launch Singapore provide, buyers might be spoilt for alternative and this in flip will lead to their reluctance to pay a premium for potential models. The complete textual content of the Copyright Act (Cap sixty three) and different statutes regarding IPR might be found on the Singapore Statutes Online Website online The Group's income jumped forty.1 p.c to $324.5 million from $231.6 million in FY 2013, lifted by increased development income and sales of growth properties in Singapore and China. Actual Estate Shopping for
  
  One factor we've on this nation is a big group of "economists," and "market analysts." What's interesting about this group of real property market-watchers is that there are two very other ways wherein they predict Boomers will affect housing markets over the subsequent decade. Let's check out those two opposites and see how every can change the best way real property investors strategy their markets. The good news is that actual property buyers are prepared for either state of affairs, and there's profit in being ready. I'm excited and searching ahead to the alternatives both or each of these conditions will supply; thank you Boomers! Mapletree to further broaden past Asia Why fortune will favour the brave in Asia's closing real property frontier
  
  The story of the 23.2 home begins with a stack of Douglas fir beams salvaged from varied demolished warehouses owned by the consumer's household for a number of generations. Design and structure innovator Omer Arbel, configured them to type a triangulated roof, which makes up one of the placing features of the home. The transient from the entrepreneur-proprietor was not solely to design a house that integrates antique wood beams, however one which erases the excellence between inside and exterior. Built on a gentle slope on a large rural acreage surrounded by two masses of previous-development forests, the indoors movement seamlessly to the outdoors and, from the within looking, one enjoys unobstructed views of the existing natural panorama which is preserved
  
  First, there are typically extra rental transactions than gross sales transactions, to permit AV to be decided for each property primarily based on comparable properties. Second, movements in sale costs are more unstable than leases. Hence, utilizing rental transactions to derive the AV helps to maintain property tax more steady for property homeowners. If you're buying or trying to lease a property. It's tiring to call up individual property agent, organize appointments, coordinate timing and to go for particular person property viewing. What most individuals do is to have a property agent representing them who will arrange and coordinate the viewings for all the properties out there based mostly on your necessities & most well-liked timing. Rent Property District 12 Rent Property District thirteen
  
  The Annual Worth of a property is mostly derived based mostly on the estimated annual hire that it may well fetch if it have been rented out. In determining the Annual Worth of a property, IRAS will think about the leases of similar properties within the vicinity, dimension and condition of the property, and different relevant components. The Annual Worth of a property is determined in the identical method regardless of whether the property is let-out, proprietor-occupied or vacant. The Annual Worth of land is determined at 5% of the market price of the land. When a constructing is demolished, the Annual Worth of the land is assessed by this method. Property Tax on Residential Properties Buyer Stamp Responsibility on Buy of Properties – Business and residential properties Rent House District 01
  
  Within the event the Bank's valuation is decrease than the acquisition price, the purchaser has to pay the distinction between the purchase value and the Bank's valuation utilizing money. As such, the money required up-front might be increased so it's at all times essential to know the valuation of the property before making any offer. Appoint Lawyer The Bank will prepare for a proper valuation of the property by way of physical inspection The completion statement will present you the balance of the acquisition price that you must pay after deducting any deposit, pro-rated property tax and utility costs, upkeep prices, and different relevant expenses in addition to any fees payable to the agent and the lawyer. Stamp Responsibility Primarily based on the Purchase Price or Market Value, whichever is larger

• The product of the areas of the two triangles formed by one diagonal equals the product of the areas of the two triangles formed by the other diagonal.Primarily based on the most recent URA personal property value index (PPPI) flash estimates, we know that the PPPI, which represents the overall real property price development, has dipped in 2013Q4. That is the first dip the market has seen within the final 2 years.
  
  To give you some perspective, the entire number of personal properties in Singapore (together with govt condominiums) is 297,689 in 2013Q3. Primarily based on the projection that there will be 19,302 units accomplished in 2014, the rise in residential models works out to be more than 6%. With a lot New Ec Launch Singapore provide, buyers might be spoilt for alternative and this in flip will lead to their reluctance to pay a premium for potential models. The complete textual content of the Copyright Act (Cap sixty three) and different statutes regarding IPR might be found on the Singapore Statutes Online Website online The Group's income jumped forty.1 p.c to $324.5 million from $231.6 million in FY 2013, lifted by increased development income and sales of growth properties in Singapore and China. Actual Estate Shopping for
  
  One factor we've on this nation is a big group of "economists," and "market analysts." What's interesting about this group of real property market-watchers is that there are two very other ways wherein they predict Boomers will affect housing markets over the subsequent decade. Let's check out those two opposites and see how every can change the best way real property investors strategy their markets. The good news is that actual property buyers are prepared for either state of affairs, and there's profit in being ready. I'm excited and searching ahead to the alternatives both or each of these conditions will supply; thank you Boomers! Mapletree to further broaden past Asia Why fortune will favour the brave in Asia's closing real property frontier
  
  The story of the 23.2 home begins with a stack of Douglas fir beams salvaged from varied demolished warehouses owned by the consumer's household for a number of generations. Design and structure innovator Omer Arbel, configured them to type a triangulated roof, which makes up one of the placing features of the home. The transient from the entrepreneur-proprietor was not solely to design a house that integrates antique wood beams, however one which erases the excellence between inside and exterior. Built on a gentle slope on a large rural acreage surrounded by two masses of previous-development forests, the indoors movement seamlessly to the outdoors and, from the within looking, one enjoys unobstructed views of the existing natural panorama which is preserved
  
  First, there are typically extra rental transactions than gross sales transactions, to permit AV to be decided for each property primarily based on comparable properties. Second, movements in sale costs are more unstable than leases. Hence, utilizing rental transactions to derive the AV helps to maintain property tax more steady for property homeowners. If you're buying or trying to lease a property. It's tiring to call up individual property agent, organize appointments, coordinate timing and to go for particular person property viewing. What most individuals do is to have a property agent representing them who will arrange and coordinate the viewings for all the properties out there based mostly on your necessities & most well-liked timing. Rent Property District 12 Rent Property District thirteen
  
  The Annual Worth of a property is mostly derived based mostly on the estimated annual hire that it may well fetch if it have been rented out. In determining the Annual Worth of a property, IRAS will think about the leases of similar properties within the vicinity, dimension and condition of the property, and different relevant components. The Annual Worth of a property is determined in the identical method regardless of whether the property is let-out, proprietor-occupied or vacant. The Annual Worth of land is determined at 5% of the market price of the land. When a constructing is demolished, the Annual Worth of the land is assessed by this method. Property Tax on Residential Properties Buyer Stamp Responsibility on Buy of Properties – Business and residential properties Rent House District 01
  
  Within the event the Bank's valuation is decrease than the acquisition price, the purchaser has to pay the distinction between the purchase value and the Bank's valuation utilizing money. As such, the money required up-front might be increased so it's at all times essential to know the valuation of the property before making any offer. Appoint Lawyer The Bank will prepare for a proper valuation of the property by way of physical inspection The completion statement will present you the balance of the acquisition price that you must pay after deducting any deposit, pro-rated property tax and utility costs, upkeep prices, and different relevant expenses in addition to any fees payable to the agent and the lawyer. Stamp Responsibility Primarily based on the Purchase Price or Market Value, whichever is larger

• The areas S and T of some two opposite triangles of the four triangles formed by the diagonals verify the equation

${\sqrt {K}}={\sqrt {S}}+{\sqrt {T}},$

where K is the area of the quadrilateral.Primarily based on the most recent URA personal property value index (PPPI) flash estimates, we know that the PPPI, which represents the overall real property price development, has dipped in 2013Q4. That is the first dip the market has seen within the final 2 years.

To give you some perspective, the entire number of personal properties in Singapore (together with govt condominiums) is 297,689 in 2013Q3. Primarily based on the projection that there will be 19,302 units accomplished in 2014, the rise in residential models works out to be more than 6%. With a lot New Ec Launch Singapore provide, buyers might be spoilt for alternative and this in flip will lead to their reluctance to pay a premium for potential models. The complete textual content of the Copyright Act (Cap sixty three) and different statutes regarding IPR might be found on the Singapore Statutes Online Website online The Group's income jumped forty.1 p.c to $324.5 million from $231.6 million in FY 2013, lifted by increased development income and sales of growth properties in Singapore and China. Actual Estate Shopping for

One factor we've on this nation is a big group of "economists," and "market analysts." What's interesting about this group of real property market-watchers is that there are two very other ways wherein they predict Boomers will affect housing markets over the subsequent decade. Let's check out those two opposites and see how every can change the best way real property investors strategy their markets. The good news is that actual property buyers are prepared for either state of affairs, and there's profit in being ready. I'm excited and searching ahead to the alternatives both or each of these conditions will supply; thank you Boomers! Mapletree to further broaden past Asia Why fortune will favour the brave in Asia's closing real property frontier

The story of the 23.2 home begins with a stack of Douglas fir beams salvaged from varied demolished warehouses owned by the consumer's household for a number of generations. Design and structure innovator Omer Arbel, configured them to type a triangulated roof, which makes up one of the placing features of the home. The transient from the entrepreneur-proprietor was not solely to design a house that integrates antique wood beams, however one which erases the excellence between inside and exterior. Built on a gentle slope on a large rural acreage surrounded by two masses of previous-development forests, the indoors movement seamlessly to the outdoors and, from the within looking, one enjoys unobstructed views of the existing natural panorama which is preserved

First, there are typically extra rental transactions than gross sales transactions, to permit AV to be decided for each property primarily based on comparable properties. Second, movements in sale costs are more unstable than leases. Hence, utilizing rental transactions to derive the AV helps to maintain property tax more steady for property homeowners. If you're buying or trying to lease a property. It's tiring to call up individual property agent, organize appointments, coordinate timing and to go for particular person property viewing. What most individuals do is to have a property agent representing them who will arrange and coordinate the viewings for all the properties out there based mostly on your necessities & most well-liked timing. Rent Property District 12 Rent Property District thirteen

The Annual Worth of a property is mostly derived based mostly on the estimated annual hire that it may well fetch if it have been rented out. In determining the Annual Worth of a property, IRAS will think about the leases of similar properties within the vicinity, dimension and condition of the property, and different relevant components. The Annual Worth of a property is determined in the identical method regardless of whether the property is let-out, proprietor-occupied or vacant. The Annual Worth of land is determined at 5% of the market price of the land. When a constructing is demolished, the Annual Worth of the land is assessed by this method. Property Tax on Residential Properties Buyer Stamp Responsibility on Buy of Properties – Business and residential properties Rent House District 01

Within the event the Bank's valuation is decrease than the acquisition price, the purchaser has to pay the distinction between the purchase value and the Bank's valuation utilizing money. As such, the money required up-front might be increased so it's at all times essential to know the valuation of the property before making any offer. Appoint Lawyer The Bank will prepare for a proper valuation of the property by way of physical inspection The completion statement will present you the balance of the acquisition price that you must pay after deducting any deposit, pro-rated property tax and utility costs, upkeep prices, and different relevant expenses in addition to any fees payable to the agent and the lawyer. Stamp Responsibility Primarily based on the Purchase Price or Market Value, whichever is larger

• The midpoints of two opposite sides and the intersection of the diagonals are collinear.Primarily based on the most recent URA personal property value index (PPPI) flash estimates, we know that the PPPI, which represents the overall real property price development, has dipped in 2013Q4. That is the first dip the market has seen within the final 2 years.
  
  To give you some perspective, the entire number of personal properties in Singapore (together with govt condominiums) is 297,689 in 2013Q3. Primarily based on the projection that there will be 19,302 units accomplished in 2014, the rise in residential models works out to be more than 6%. With a lot New Ec Launch Singapore provide, buyers might be spoilt for alternative and this in flip will lead to their reluctance to pay a premium for potential models. The complete textual content of the Copyright Act (Cap sixty three) and different statutes regarding IPR might be found on the Singapore Statutes Online Website online The Group's income jumped forty.1 p.c to $324.5 million from $231.6 million in FY 2013, lifted by increased development income and sales of growth properties in Singapore and China. Actual Estate Shopping for
  
  One factor we've on this nation is a big group of "economists," and "market analysts." What's interesting about this group of real property market-watchers is that there are two very other ways wherein they predict Boomers will affect housing markets over the subsequent decade. Let's check out those two opposites and see how every can change the best way real property investors strategy their markets. The good news is that actual property buyers are prepared for either state of affairs, and there's profit in being ready. I'm excited and searching ahead to the alternatives both or each of these conditions will supply; thank you Boomers! Mapletree to further broaden past Asia Why fortune will favour the brave in Asia's closing real property frontier
  
  The story of the 23.2 home begins with a stack of Douglas fir beams salvaged from varied demolished warehouses owned by the consumer's household for a number of generations. Design and structure innovator Omer Arbel, configured them to type a triangulated roof, which makes up one of the placing features of the home. The transient from the entrepreneur-proprietor was not solely to design a house that integrates antique wood beams, however one which erases the excellence between inside and exterior. Built on a gentle slope on a large rural acreage surrounded by two masses of previous-development forests, the indoors movement seamlessly to the outdoors and, from the within looking, one enjoys unobstructed views of the existing natural panorama which is preserved
  
  First, there are typically extra rental transactions than gross sales transactions, to permit AV to be decided for each property primarily based on comparable properties. Second, movements in sale costs are more unstable than leases. Hence, utilizing rental transactions to derive the AV helps to maintain property tax more steady for property homeowners. If you're buying or trying to lease a property. It's tiring to call up individual property agent, organize appointments, coordinate timing and to go for particular person property viewing. What most individuals do is to have a property agent representing them who will arrange and coordinate the viewings for all the properties out there based mostly on your necessities & most well-liked timing. Rent Property District 12 Rent Property District thirteen
  
  The Annual Worth of a property is mostly derived based mostly on the estimated annual hire that it may well fetch if it have been rented out. In determining the Annual Worth of a property, IRAS will think about the leases of similar properties within the vicinity, dimension and condition of the property, and different relevant components. The Annual Worth of a property is determined in the identical method regardless of whether the property is let-out, proprietor-occupied or vacant. The Annual Worth of land is determined at 5% of the market price of the land. When a constructing is demolished, the Annual Worth of the land is assessed by this method. Property Tax on Residential Properties Buyer Stamp Responsibility on Buy of Properties – Business and residential properties Rent House District 01
  
  Within the event the Bank's valuation is decrease than the acquisition price, the purchaser has to pay the distinction between the purchase value and the Bank's valuation utilizing money. As such, the money required up-front might be increased so it's at all times essential to know the valuation of the property before making any offer. Appoint Lawyer The Bank will prepare for a proper valuation of the property by way of physical inspection The completion statement will present you the balance of the acquisition price that you must pay after deducting any deposit, pro-rated property tax and utility costs, upkeep prices, and different relevant expenses in addition to any fees payable to the agent and the lawyer. Stamp Responsibility Primarily based on the Purchase Price or Market Value, whichever is larger

Additionally, the following properties are equivalent, and each implies that opposite sides a and b are parallel:

• The consecutive sides a, c, b, d and the diagonals p, q verify the equation Primarily based on the most recent URA personal property value index (PPPI) flash estimates, we know that the PPPI, which represents the overall real property price development, has dipped in 2013Q4. That is the first dip the market has seen within the final 2 years.
  
  To give you some perspective, the entire number of personal properties in Singapore (together with govt condominiums) is 297,689 in 2013Q3. Primarily based on the projection that there will be 19,302 units accomplished in 2014, the rise in residential models works out to be more than 6%. With a lot New Ec Launch Singapore provide, buyers might be spoilt for alternative and this in flip will lead to their reluctance to pay a premium for potential models. The complete textual content of the Copyright Act (Cap sixty three) and different statutes regarding IPR might be found on the Singapore Statutes Online Website online The Group's income jumped forty.1 p.c to $324.5 million from $231.6 million in FY 2013, lifted by increased development income and sales of growth properties in Singapore and China. Actual Estate Shopping for
  
  One factor we've on this nation is a big group of "economists," and "market analysts." What's interesting about this group of real property market-watchers is that there are two very other ways wherein they predict Boomers will affect housing markets over the subsequent decade. Let's check out those two opposites and see how every can change the best way real property investors strategy their markets. The good news is that actual property buyers are prepared for either state of affairs, and there's profit in being ready. I'm excited and searching ahead to the alternatives both or each of these conditions will supply; thank you Boomers! Mapletree to further broaden past Asia Why fortune will favour the brave in Asia's closing real property frontier
  
  The story of the 23.2 home begins with a stack of Douglas fir beams salvaged from varied demolished warehouses owned by the consumer's household for a number of generations. Design and structure innovator Omer Arbel, configured them to type a triangulated roof, which makes up one of the placing features of the home. The transient from the entrepreneur-proprietor was not solely to design a house that integrates antique wood beams, however one which erases the excellence between inside and exterior. Built on a gentle slope on a large rural acreage surrounded by two masses of previous-development forests, the indoors movement seamlessly to the outdoors and, from the within looking, one enjoys unobstructed views of the existing natural panorama which is preserved
  
  First, there are typically extra rental transactions than gross sales transactions, to permit AV to be decided for each property primarily based on comparable properties. Second, movements in sale costs are more unstable than leases. Hence, utilizing rental transactions to derive the AV helps to maintain property tax more steady for property homeowners. If you're buying or trying to lease a property. It's tiring to call up individual property agent, organize appointments, coordinate timing and to go for particular person property viewing. What most individuals do is to have a property agent representing them who will arrange and coordinate the viewings for all the properties out there based mostly on your necessities & most well-liked timing. Rent Property District 12 Rent Property District thirteen
  
  The Annual Worth of a property is mostly derived based mostly on the estimated annual hire that it may well fetch if it have been rented out. In determining the Annual Worth of a property, IRAS will think about the leases of similar properties within the vicinity, dimension and condition of the property, and different relevant components. The Annual Worth of a property is determined in the identical method regardless of whether the property is let-out, proprietor-occupied or vacant. The Annual Worth of land is determined at 5% of the market price of the land. When a constructing is demolished, the Annual Worth of the land is assessed by this method. Property Tax on Residential Properties Buyer Stamp Responsibility on Buy of Properties – Business and residential properties Rent House District 01
  
  Within the event the Bank's valuation is decrease than the acquisition price, the purchaser has to pay the distinction between the purchase value and the Bank's valuation utilizing money. As such, the money required up-front might be increased so it's at all times essential to know the valuation of the property before making any offer. Appoint Lawyer The Bank will prepare for a proper valuation of the property by way of physical inspection The completion statement will present you the balance of the acquisition price that you must pay after deducting any deposit, pro-rated property tax and utility costs, upkeep prices, and different relevant expenses in addition to any fees payable to the agent and the lawyer. Stamp Responsibility Primarily based on the Purchase Price or Market Value, whichever is larger

$p^{2}+q^{2}=c^{2}+d^{2}+2ab.$

• The distance v between the midpoints of the diagonals verifies the equationPrimarily based on the most recent URA personal property value index (PPPI) flash estimates, we know that the PPPI, which represents the overall real property price development, has dipped in 2013Q4. That is the first dip the market has seen within the final 2 years.
  
  To give you some perspective, the entire number of personal properties in Singapore (together with govt condominiums) is 297,689 in 2013Q3. Primarily based on the projection that there will be 19,302 units accomplished in 2014, the rise in residential models works out to be more than 6%. With a lot New Ec Launch Singapore provide, buyers might be spoilt for alternative and this in flip will lead to their reluctance to pay a premium for potential models. The complete textual content of the Copyright Act (Cap sixty three) and different statutes regarding IPR might be found on the Singapore Statutes Online Website online The Group's income jumped forty.1 p.c to $324.5 million from $231.6 million in FY 2013, lifted by increased development income and sales of growth properties in Singapore and China. Actual Estate Shopping for
  
  One factor we've on this nation is a big group of "economists," and "market analysts." What's interesting about this group of real property market-watchers is that there are two very other ways wherein they predict Boomers will affect housing markets over the subsequent decade. Let's check out those two opposites and see how every can change the best way real property investors strategy their markets. The good news is that actual property buyers are prepared for either state of affairs, and there's profit in being ready. I'm excited and searching ahead to the alternatives both or each of these conditions will supply; thank you Boomers! Mapletree to further broaden past Asia Why fortune will favour the brave in Asia's closing real property frontier
  
  The story of the 23.2 home begins with a stack of Douglas fir beams salvaged from varied demolished warehouses owned by the consumer's household for a number of generations. Design and structure innovator Omer Arbel, configured them to type a triangulated roof, which makes up one of the placing features of the home. The transient from the entrepreneur-proprietor was not solely to design a house that integrates antique wood beams, however one which erases the excellence between inside and exterior. Built on a gentle slope on a large rural acreage surrounded by two masses of previous-development forests, the indoors movement seamlessly to the outdoors and, from the within looking, one enjoys unobstructed views of the existing natural panorama which is preserved
  
  First, there are typically extra rental transactions than gross sales transactions, to permit AV to be decided for each property primarily based on comparable properties. Second, movements in sale costs are more unstable than leases. Hence, utilizing rental transactions to derive the AV helps to maintain property tax more steady for property homeowners. If you're buying or trying to lease a property. It's tiring to call up individual property agent, organize appointments, coordinate timing and to go for particular person property viewing. What most individuals do is to have a property agent representing them who will arrange and coordinate the viewings for all the properties out there based mostly on your necessities & most well-liked timing. Rent Property District 12 Rent Property District thirteen
  
  The Annual Worth of a property is mostly derived based mostly on the estimated annual hire that it may well fetch if it have been rented out. In determining the Annual Worth of a property, IRAS will think about the leases of similar properties within the vicinity, dimension and condition of the property, and different relevant components. The Annual Worth of a property is determined in the identical method regardless of whether the property is let-out, proprietor-occupied or vacant. The Annual Worth of land is determined at 5% of the market price of the land. When a constructing is demolished, the Annual Worth of the land is assessed by this method. Property Tax on Residential Properties Buyer Stamp Responsibility on Buy of Properties – Business and residential properties Rent House District 01
  
  Within the event the Bank's valuation is decrease than the acquisition price, the purchaser has to pay the distinction between the purchase value and the Bank's valuation utilizing money. As such, the money required up-front might be increased so it's at all times essential to know the valuation of the property before making any offer. Appoint Lawyer The Bank will prepare for a proper valuation of the property by way of physical inspection The completion statement will present you the balance of the acquisition price that you must pay after deducting any deposit, pro-rated property tax and utility costs, upkeep prices, and different relevant expenses in addition to any fees payable to the agent and the lawyer. Stamp Responsibility Primarily based on the Purchase Price or Market Value, whichever is larger

$v={\frac {|a-b|}{2}}.$

Midsegment and height

The midsegment (also called the median or midline) of a trapezoid is the segment that joins the midpoints of the legs. It is parallel to the bases. Its length m is equal to the average of the lengths of the bases a and b of the trapezoid,

$m={\frac {a+b}{2}}.$

The midsegment of a trapezoid is one of the two bimedians (the other bimedian divides the trapezoid into equal areas).

The height (or altitude) is the perpendicular distance between the bases. In the case that the two bases have different lengths (a ≠ b), the height of a trapezoid h can be determined by the length of its four sides using the formula

$h={\frac {\sqrt {(-a+b+c+d)(a-b+c+d)(a-b+c-d)(a-b-c+d)}}{2|b-a|}}$

where c and d are the lengths of the legs. This formula also gives a way of determining when a trapezoid of consecutive sides a, c, b, and d exists. There is such a trapezoid with bases a and b if and only if

$\displaystyle h^{2}>0.$

Area

The area K of a trapezoid is given by

$K={\frac {a+b}{2}}\cdot h$

where a and b are the lengths of the parallel sides, and h is the height (the perpendicular distance between these sides.) In 499 AD Aryabhata, a great mathematician-astronomer from the classical age of Indian mathematics and Indian astronomy, used this method in the Aryabhatiya (section 2.8). This yields as a special case the well-known formula for the area of a triangle, by considering a triangle as a degenerate trapezoid in which one of the parallel sides has shrunk to a point.

Therefore the area of a trapezoid is equal to the length of this midsegment multiplied by the height

$K=mh\,$

From the formula for the height, it can be concluded that the area can be expressed in terms of the four sides as

$K={\frac {a+b}{4|b-a|}}{\sqrt {(-a+b+c+d)(a-b+c+d)(a-b+c-d)(a-b-c+d)}}.$

When one of the parallel sides has shrunk to a point (say a = 0), this formula reduces to Heron's formula for the area of a triangle.

Another equivalent formula for the area, which more closely resembles Heron's formula, is

$K={\frac {a+b}{|b-a|}}{\sqrt {(s-b)(s-a)(s-b-c)(s-b-d)}},$

where $s={\tfrac {1}{2}}(a+b+c+d)$ is the semiperimeter of the trapezoid. (This formula is similar to Brahmagupta's formula, but it differs from it, in that a trapezoid might not be cyclic (inscribed in a circle). The formula is also a special case of Bretschneider's formula for a general quadrilateral).

From Bretschneider's formula, it follows that

$K={\sqrt {{\frac {(ab^{2}-a^{2}b-ad^{2}+bc^{2})(ab^{2}-a^{2}b-ac^{2}+bd^{2})}{(2(b-a))^{2}}}-\left({\frac {b^{2}+d^{2}-a^{2}-c^{2}}{4}}\right)^{2}}}.$

The line that joins the midpoints of the parallel sides, bisects the area.

Diagonals

The lengths of the diagonals are

$p={\sqrt {\frac {ab^{2}-a^{2}b-ac^{2}+bd^{2}}{b-a}}},$

$q={\sqrt {\frac {ab^{2}-a^{2}b-ad^{2}+bc^{2}}{b-a}}}$

where a and b are the bases, c and d are the other two sides, and a < b.

If the trapezoid is divided into four triangles by its diagonals AC and BD (as shown on the right), intersecting at O, then the area of A Quasi Fermi level (also called imref) is a term used in quantum mechanics and especially in solid state physics for the new Fermi level that describes the population of each type of charge carrier (electrons and holes) in a semiconductor separately when their populations are displaced from equilibrium. This displacement could be caused by the application of an external electric potential, which injects electrons and holes, or by exposure to light of energy $E>E_{g}$, which increases the density of both electrons and holes above their equilibrium values. The displacement from equilibrium is such that the carrier populations can no longer be described by a single Fermi level, and separate quasi-Fermi levels must be used for each charge carrier.

(In this article the term "Fermi level" is used to mean the same thing as chemical potential. This is the most common usage in semiconductor physics, although in other areas of solid-state physics the term "Fermi level" means something different, namely the chemical potential at zero temperature.)

When a semiconductor is in thermal equilibrium, the distribution function of the electrons at the energy level of E is presented by Fermi-Dirac distribution function. In this case the Fermi level is defined as the level in which the probability of occupation of electron at that energy is 1/2. In thermal equilibrium, the electron quasi-Fermi level, hole quasi-Fermi level and Fermi level are equal.

PN junction operation in forward bias mode showing reducing depletion width. Both p and n junctions are doped at a 1e15/cm3 doping level, leading to built-in potential of ~0.59V. Observe the different quasi-fermi levels for electrons and holes in n and p regions (red curves).

When a disturbance from a thermal equilibrium situation occurs, the populations of holes and electrons change. If the disturbance is not too great or not changing too quickly, the populations of electrons and holes each relax to a state of quasi thermal equilibrium. Because the relaxation time for electrons within the conduction band is much lower than across the band gap, we can consider that the electrons are in thermal equilibrium in the conduction band. This is also applicable for holes in the valence band. In the case of electrons we can define a quasi Fermi level and temperature due to thermal equilibrium of electrons in conduction band and quasi Fermi level for holes similarly.

We can state the general Fermi function for electrons in conduction band as

$f_{c}(k,r)\approx f_{0}(E,E_{Fn},T_{n})$

and for holes in valence band as

$f_{v}(k,r)\approx f_{0}(E,E_{Fp},T_{p})$

where:

• $f_{0}(E,E_{F},T)={\frac {1}{1+e^{(E-E_{F})/(k_{B}T)}}}$ is the Fermi-Dirac distribution function,
• $E_{Fn}$ is the electron quasi-fermi level at location r,
• $E_{Fp}$ is the hole quasi-fermi level at location r,
• $T_{n}$ is the electron temperature,
• $T_{p}$ is the hole temperature,
• $f_{c}(k,r)$ is the probability that a particular conduction-band state, with wavevector k and position r, is occupied by an electron,
• $f_{v}(k,r)$ is the probability that a particular valence-band state, with wavevector k and position r, is occupied by an electron, i.e. not occupied by a hole,
• $E$ is the energy of the conduction- or valence-band state in question,
• $k_{B}$ is Boltzmann's constant.

This simplification will help us in many areas. For example, we can use the same equation for electron and hole densities used in thermal equilibrium, but substituting the Fermi levels and temperature. i.e.:

$n=n_{i}(T_{n})e^{(E_{Fn}-E_{i})/(k_{B}T_{n})}$

and

$p=p_{i}(T_{p})e^{(E_{i}-E_{Fp})/(k_{B}T_{p})}$

where

• $n$ is the density (number per unit volume) of conduction electrons,
• $p$ is the density of holes,
• $n_{i}(T_{n})$ is the density of conduction electrons that would be present in thermal equilibrium if the Fermi level were at $E_{i}$ and the temperature were $T_{n}$
• $p_{i}(T_{p})$ is the density of holes that would be present in thermal equilibrium if the Fermi level were at $E_{i}$ and the temperature were $T_{p}$,
• This equation is valid in the Boltzmann approximation, i.e. assuming the electron and hole densities are not too high.

A current (due to the combined effects of drift and diffusion) will only flow if there is a variation in the Fermi or quasi Fermi level. The current density for electron flow can be shown to be proportional to the gradient in the electron quasi Fermi level as such:

$J_{n}(r)=\mu _{n}n\nabla _{r}(E_{Fn})$, where $\mu$ is the electron mobility, and $\nabla _{r}$ is the gradient with respect to distance r.

Similarly, for holes, we have

$J_{p}(r)=\mu _{p}p\nabla _{r}(E_{Fp}).$

Notes

Nelson, Jenny. The Physics of Solar Cells. Imperial College Press, 2003. is equal to that of A Quasi Fermi level (also called imref) is a term used in quantum mechanics and especially in solid state physics for the new Fermi level that describes the population of each type of charge carrier (electrons and holes) in a semiconductor separately when their populations are displaced from equilibrium. This displacement could be caused by the application of an external electric potential, which injects electrons and holes, or by exposure to light of energy $E>E_{g}$, which increases the density of both electrons and holes above their equilibrium values. The displacement from equilibrium is such that the carrier populations can no longer be described by a single Fermi level, and separate quasi-Fermi levels must be used for each charge carrier.

(In this article the term "Fermi level" is used to mean the same thing as chemical potential. This is the most common usage in semiconductor physics, although in other areas of solid-state physics the term "Fermi level" means something different, namely the chemical potential at zero temperature.)

When a semiconductor is in thermal equilibrium, the distribution function of the electrons at the energy level of E is presented by Fermi-Dirac distribution function. In this case the Fermi level is defined as the level in which the probability of occupation of electron at that energy is 1/2. In thermal equilibrium, the electron quasi-Fermi level, hole quasi-Fermi level and Fermi level are equal.

PN junction operation in forward bias mode showing reducing depletion width. Both p and n junctions are doped at a 1e15/cm3 doping level, leading to built-in potential of ~0.59V. Observe the different quasi-fermi levels for electrons and holes in n and p regions (red curves).

When a disturbance from a thermal equilibrium situation occurs, the populations of holes and electrons change. If the disturbance is not too great or not changing too quickly, the populations of electrons and holes each relax to a state of quasi thermal equilibrium. Because the relaxation time for electrons within the conduction band is much lower than across the band gap, we can consider that the electrons are in thermal equilibrium in the conduction band. This is also applicable for holes in the valence band. In the case of electrons we can define a quasi Fermi level and temperature due to thermal equilibrium of electrons in conduction band and quasi Fermi level for holes similarly.

We can state the general Fermi function for electrons in conduction band as

$f_{c}(k,r)\approx f_{0}(E,E_{Fn},T_{n})$

and for holes in valence band as

$f_{v}(k,r)\approx f_{0}(E,E_{Fp},T_{p})$

where:

• $f_{0}(E,E_{F},T)={\frac {1}{1+e^{(E-E_{F})/(k_{B}T)}}}$ is the Fermi-Dirac distribution function,
• $E_{Fn}$ is the electron quasi-fermi level at location r,
• $E_{Fp}$ is the hole quasi-fermi level at location r,
• $T_{n}$ is the electron temperature,
• $T_{p}$ is the hole temperature,
• $f_{c}(k,r)$ is the probability that a particular conduction-band state, with wavevector k and position r, is occupied by an electron,
• $f_{v}(k,r)$ is the probability that a particular valence-band state, with wavevector k and position r, is occupied by an electron, i.e. not occupied by a hole,
• $E$ is the energy of the conduction- or valence-band state in question,
• $k_{B}$ is Boltzmann's constant.

This simplification will help us in many areas. For example, we can use the same equation for electron and hole densities used in thermal equilibrium, but substituting the Fermi levels and temperature. i.e.:

$n=n_{i}(T_{n})e^{(E_{Fn}-E_{i})/(k_{B}T_{n})}$

and

$p=p_{i}(T_{p})e^{(E_{i}-E_{Fp})/(k_{B}T_{p})}$

where

• $n$ is the density (number per unit volume) of conduction electrons,
• $p$ is the density of holes,
• $n_{i}(T_{n})$ is the density of conduction electrons that would be present in thermal equilibrium if the Fermi level were at $E_{i}$ and the temperature were $T_{n}$
• $p_{i}(T_{p})$ is the density of holes that would be present in thermal equilibrium if the Fermi level were at $E_{i}$ and the temperature were $T_{p}$,
• This equation is valid in the Boltzmann approximation, i.e. assuming the electron and hole densities are not too high.

A current (due to the combined effects of drift and diffusion) will only flow if there is a variation in the Fermi or quasi Fermi level. The current density for electron flow can be shown to be proportional to the gradient in the electron quasi Fermi level as such:

$J_{n}(r)=\mu _{n}n\nabla _{r}(E_{Fn})$, where $\mu$ is the electron mobility, and $\nabla _{r}$ is the gradient with respect to distance r.

Similarly, for holes, we have

$J_{p}(r)=\mu _{p}p\nabla _{r}(E_{Fp}).$

Notes

Nelson, Jenny. The Physics of Solar Cells. Imperial College Press, 2003., and the product of the areas of A Quasi Fermi level (also called imref) is a term used in quantum mechanics and especially in solid state physics for the new Fermi level that describes the population of each type of charge carrier (electrons and holes) in a semiconductor separately when their populations are displaced from equilibrium. This displacement could be caused by the application of an external electric potential, which injects electrons and holes, or by exposure to light of energy $E>E_{g}$, which increases the density of both electrons and holes above their equilibrium values. The displacement from equilibrium is such that the carrier populations can no longer be described by a single Fermi level, and separate quasi-Fermi levels must be used for each charge carrier.

(In this article the term "Fermi level" is used to mean the same thing as chemical potential. This is the most common usage in semiconductor physics, although in other areas of solid-state physics the term "Fermi level" means something different, namely the chemical potential at zero temperature.)

When a semiconductor is in thermal equilibrium, the distribution function of the electrons at the energy level of E is presented by Fermi-Dirac distribution function. In this case the Fermi level is defined as the level in which the probability of occupation of electron at that energy is 1/2. In thermal equilibrium, the electron quasi-Fermi level, hole quasi-Fermi level and Fermi level are equal.

PN junction operation in forward bias mode showing reducing depletion width. Both p and n junctions are doped at a 1e15/cm3 doping level, leading to built-in potential of ~0.59V. Observe the different quasi-fermi levels for electrons and holes in n and p regions (red curves).

When a disturbance from a thermal equilibrium situation occurs, the populations of holes and electrons change. If the disturbance is not too great or not changing too quickly, the populations of electrons and holes each relax to a state of quasi thermal equilibrium. Because the relaxation time for electrons within the conduction band is much lower than across the band gap, we can consider that the electrons are in thermal equilibrium in the conduction band. This is also applicable for holes in the valence band. In the case of electrons we can define a quasi Fermi level and temperature due to thermal equilibrium of electrons in conduction band and quasi Fermi level for holes similarly.

We can state the general Fermi function for electrons in conduction band as

$f_{c}(k,r)\approx f_{0}(E,E_{Fn},T_{n})$

and for holes in valence band as

$f_{v}(k,r)\approx f_{0}(E,E_{Fp},T_{p})$

where:

• $f_{0}(E,E_{F},T)={\frac {1}{1+e^{(E-E_{F})/(k_{B}T)}}}$ is the Fermi-Dirac distribution function,
• $E_{Fn}$ is the electron quasi-fermi level at location r,
• $E_{Fp}$ is the hole quasi-fermi level at location r,
• $T_{n}$ is the electron temperature,
• $T_{p}$ is the hole temperature,
• $f_{c}(k,r)$ is the probability that a particular conduction-band state, with wavevector k and position r, is occupied by an electron,
• $f_{v}(k,r)$ is the probability that a particular valence-band state, with wavevector k and position r, is occupied by an electron, i.e. not occupied by a hole,
• $E$ is the energy of the conduction- or valence-band state in question,
• $k_{B}$ is Boltzmann's constant.

This simplification will help us in many areas. For example, we can use the same equation for electron and hole densities used in thermal equilibrium, but substituting the Fermi levels and temperature. i.e.:

$n=n_{i}(T_{n})e^{(E_{Fn}-E_{i})/(k_{B}T_{n})}$

and

$p=p_{i}(T_{p})e^{(E_{i}-E_{Fp})/(k_{B}T_{p})}$

where

• $n$ is the density (number per unit volume) of conduction electrons,
• $p$ is the density of holes,
• $n_{i}(T_{n})$ is the density of conduction electrons that would be present in thermal equilibrium if the Fermi level were at $E_{i}$ and the temperature were $T_{n}$
• $p_{i}(T_{p})$ is the density of holes that would be present in thermal equilibrium if the Fermi level were at $E_{i}$ and the temperature were $T_{p}$,
• This equation is valid in the Boltzmann approximation, i.e. assuming the electron and hole densities are not too high.

A current (due to the combined effects of drift and diffusion) will only flow if there is a variation in the Fermi or quasi Fermi level. The current density for electron flow can be shown to be proportional to the gradient in the electron quasi Fermi level as such:

$J_{n}(r)=\mu _{n}n\nabla _{r}(E_{Fn})$, where $\mu$ is the electron mobility, and $\nabla _{r}$ is the gradient with respect to distance r.

Similarly, for holes, we have

$J_{p}(r)=\mu _{p}p\nabla _{r}(E_{Fp}).$

Notes

Nelson, Jenny. The Physics of Solar Cells. Imperial College Press, 2003. and A Quasi Fermi level (also called imref) is a term used in quantum mechanics and especially in solid state physics for the new Fermi level that describes the population of each type of charge carrier (electrons and holes) in a semiconductor separately when their populations are displaced from equilibrium. This displacement could be caused by the application of an external electric potential, which injects electrons and holes, or by exposure to light of energy $E>E_{g}$, which increases the density of both electrons and holes above their equilibrium values. The displacement from equilibrium is such that the carrier populations can no longer be described by a single Fermi level, and separate quasi-Fermi levels must be used for each charge carrier.

(In this article the term "Fermi level" is used to mean the same thing as chemical potential. This is the most common usage in semiconductor physics, although in other areas of solid-state physics the term "Fermi level" means something different, namely the chemical potential at zero temperature.)

When a semiconductor is in thermal equilibrium, the distribution function of the electrons at the energy level of E is presented by Fermi-Dirac distribution function. In this case the Fermi level is defined as the level in which the probability of occupation of electron at that energy is 1/2. In thermal equilibrium, the electron quasi-Fermi level, hole quasi-Fermi level and Fermi level are equal.

PN junction operation in forward bias mode showing reducing depletion width. Both p and n junctions are doped at a 1e15/cm3 doping level, leading to built-in potential of ~0.59V. Observe the different quasi-fermi levels for electrons and holes in n and p regions (red curves).

When a disturbance from a thermal equilibrium situation occurs, the populations of holes and electrons change. If the disturbance is not too great or not changing too quickly, the populations of electrons and holes each relax to a state of quasi thermal equilibrium. Because the relaxation time for electrons within the conduction band is much lower than across the band gap, we can consider that the electrons are in thermal equilibrium in the conduction band. This is also applicable for holes in the valence band. In the case of electrons we can define a quasi Fermi level and temperature due to thermal equilibrium of electrons in conduction band and quasi Fermi level for holes similarly.

We can state the general Fermi function for electrons in conduction band as

$f_{c}(k,r)\approx f_{0}(E,E_{Fn},T_{n})$

and for holes in valence band as

$f_{v}(k,r)\approx f_{0}(E,E_{Fp},T_{p})$

where:

• $f_{0}(E,E_{F},T)={\frac {1}{1+e^{(E-E_{F})/(k_{B}T)}}}$ is the Fermi-Dirac distribution function,
• $E_{Fn}$ is the electron quasi-fermi level at location r,
• $E_{Fp}$ is the hole quasi-fermi level at location r,
• $T_{n}$ is the electron temperature,
• $T_{p}$ is the hole temperature,
• $f_{c}(k,r)$ is the probability that a particular conduction-band state, with wavevector k and position r, is occupied by an electron,
• $f_{v}(k,r)$ is the probability that a particular valence-band state, with wavevector k and position r, is occupied by an electron, i.e. not occupied by a hole,
• $E$ is the energy of the conduction- or valence-band state in question,
• $k_{B}$ is Boltzmann's constant.

This simplification will help us in many areas. For example, we can use the same equation for electron and hole densities used in thermal equilibrium, but substituting the Fermi levels and temperature. i.e.:

$n=n_{i}(T_{n})e^{(E_{Fn}-E_{i})/(k_{B}T_{n})}$

and

$p=p_{i}(T_{p})e^{(E_{i}-E_{Fp})/(k_{B}T_{p})}$

where

• $n$ is the density (number per unit volume) of conduction electrons,
• $p$ is the density of holes,
• $n_{i}(T_{n})$ is the density of conduction electrons that would be present in thermal equilibrium if the Fermi level were at $E_{i}$ and the temperature were $T_{n}$
• $p_{i}(T_{p})$ is the density of holes that would be present in thermal equilibrium if the Fermi level were at $E_{i}$ and the temperature were $T_{p}$,
• This equation is valid in the Boltzmann approximation, i.e. assuming the electron and hole densities are not too high.

A current (due to the combined effects of drift and diffusion) will only flow if there is a variation in the Fermi or quasi Fermi level. The current density for electron flow can be shown to be proportional to the gradient in the electron quasi Fermi level as such:

$J_{n}(r)=\mu _{n}n\nabla _{r}(E_{Fn})$, where $\mu$ is the electron mobility, and $\nabla _{r}$ is the gradient with respect to distance r.

Similarly, for holes, we have

$J_{p}(r)=\mu _{p}p\nabla _{r}(E_{Fp}).$

Notes

Nelson, Jenny. The Physics of Solar Cells. Imperial College Press, 2003. is equal to that of A Quasi Fermi level (also called imref) is a term used in quantum mechanics and especially in solid state physics for the new Fermi level that describes the population of each type of charge carrier (electrons and holes) in a semiconductor separately when their populations are displaced from equilibrium. This displacement could be caused by the application of an external electric potential, which injects electrons and holes, or by exposure to light of energy $E>E_{g}$, which increases the density of both electrons and holes above their equilibrium values. The displacement from equilibrium is such that the carrier populations can no longer be described by a single Fermi level, and separate quasi-Fermi levels must be used for each charge carrier.

(In this article the term "Fermi level" is used to mean the same thing as chemical potential. This is the most common usage in semiconductor physics, although in other areas of solid-state physics the term "Fermi level" means something different, namely the chemical potential at zero temperature.)

When a semiconductor is in thermal equilibrium, the distribution function of the electrons at the energy level of E is presented by Fermi-Dirac distribution function. In this case the Fermi level is defined as the level in which the probability of occupation of electron at that energy is 1/2. In thermal equilibrium, the electron quasi-Fermi level, hole quasi-Fermi level and Fermi level are equal.

PN junction operation in forward bias mode showing reducing depletion width. Both p and n junctions are doped at a 1e15/cm3 doping level, leading to built-in potential of ~0.59V. Observe the different quasi-fermi levels for electrons and holes in n and p regions (red curves).

When a disturbance from a thermal equilibrium situation occurs, the populations of holes and electrons change. If the disturbance is not too great or not changing too quickly, the populations of electrons and holes each relax to a state of quasi thermal equilibrium. Because the relaxation time for electrons within the conduction band is much lower than across the band gap, we can consider that the electrons are in thermal equilibrium in the conduction band. This is also applicable for holes in the valence band. In the case of electrons we can define a quasi Fermi level and temperature due to thermal equilibrium of electrons in conduction band and quasi Fermi level for holes similarly.

We can state the general Fermi function for electrons in conduction band as

$f_{c}(k,r)\approx f_{0}(E,E_{Fn},T_{n})$

and for holes in valence band as

$f_{v}(k,r)\approx f_{0}(E,E_{Fp},T_{p})$

where:

• $f_{0}(E,E_{F},T)={\frac {1}{1+e^{(E-E_{F})/(k_{B}T)}}}$ is the Fermi-Dirac distribution function,
• $E_{Fn}$ is the electron quasi-fermi level at location r,
• $E_{Fp}$ is the hole quasi-fermi level at location r,
• $T_{n}$ is the electron temperature,
• $T_{p}$ is the hole temperature,
• $f_{c}(k,r)$ is the probability that a particular conduction-band state, with wavevector k and position r, is occupied by an electron,
• $f_{v}(k,r)$ is the probability that a particular valence-band state, with wavevector k and position r, is occupied by an electron, i.e. not occupied by a hole,
• $E$ is the energy of the conduction- or valence-band state in question,
• $k_{B}$ is Boltzmann's constant.

This simplification will help us in many areas. For example, we can use the same equation for electron and hole densities used in thermal equilibrium, but substituting the Fermi levels and temperature. i.e.:

$n=n_{i}(T_{n})e^{(E_{Fn}-E_{i})/(k_{B}T_{n})}$

and

$p=p_{i}(T_{p})e^{(E_{i}-E_{Fp})/(k_{B}T_{p})}$

where

• $n$ is the density (number per unit volume) of conduction electrons,
• $p$ is the density of holes,
• $n_{i}(T_{n})$ is the density of conduction electrons that would be present in thermal equilibrium if the Fermi level were at $E_{i}$ and the temperature were $T_{n}$
• $p_{i}(T_{p})$ is the density of holes that would be present in thermal equilibrium if the Fermi level were at $E_{i}$ and the temperature were $T_{p}$,
• This equation is valid in the Boltzmann approximation, i.e. assuming the electron and hole densities are not too high.

A current (due to the combined effects of drift and diffusion) will only flow if there is a variation in the Fermi or quasi Fermi level. The current density for electron flow can be shown to be proportional to the gradient in the electron quasi Fermi level as such:

$J_{n}(r)=\mu _{n}n\nabla _{r}(E_{Fn})$, where $\mu$ is the electron mobility, and $\nabla _{r}$ is the gradient with respect to distance r.

Similarly, for holes, we have

$J_{p}(r)=\mu _{p}p\nabla _{r}(E_{Fp}).$

Notes

Nelson, Jenny. The Physics of Solar Cells. Imperial College Press, 2003. and A Quasi Fermi level (also called imref) is a term used in quantum mechanics and especially in solid state physics for the new Fermi level that describes the population of each type of charge carrier (electrons and holes) in a semiconductor separately when their populations are displaced from equilibrium. This displacement could be caused by the application of an external electric potential, which injects electrons and holes, or by exposure to light of energy $E>E_{g}$, which increases the density of both electrons and holes above their equilibrium values. The displacement from equilibrium is such that the carrier populations can no longer be described by a single Fermi level, and separate quasi-Fermi levels must be used for each charge carrier.

(In this article the term "Fermi level" is used to mean the same thing as chemical potential. This is the most common usage in semiconductor physics, although in other areas of solid-state physics the term "Fermi level" means something different, namely the chemical potential at zero temperature.)

When a semiconductor is in thermal equilibrium, the distribution function of the electrons at the energy level of E is presented by Fermi-Dirac distribution function. In this case the Fermi level is defined as the level in which the probability of occupation of electron at that energy is 1/2. In thermal equilibrium, the electron quasi-Fermi level, hole quasi-Fermi level and Fermi level are equal.

PN junction operation in forward bias mode showing reducing depletion width. Both p and n junctions are doped at a 1e15/cm3 doping level, leading to built-in potential of ~0.59V. Observe the different quasi-fermi levels for electrons and holes in n and p regions (red curves).

When a disturbance from a thermal equilibrium situation occurs, the populations of holes and electrons change. If the disturbance is not too great or not changing too quickly, the populations of electrons and holes each relax to a state of quasi thermal equilibrium. Because the relaxation time for electrons within the conduction band is much lower than across the band gap, we can consider that the electrons are in thermal equilibrium in the conduction band. This is also applicable for holes in the valence band. In the case of electrons we can define a quasi Fermi level and temperature due to thermal equilibrium of electrons in conduction band and quasi Fermi level for holes similarly.

We can state the general Fermi function for electrons in conduction band as

$f_{c}(k,r)\approx f_{0}(E,E_{Fn},T_{n})$

and for holes in valence band as

$f_{v}(k,r)\approx f_{0}(E,E_{Fp},T_{p})$

where:

• $f_{0}(E,E_{F},T)={\frac {1}{1+e^{(E-E_{F})/(k_{B}T)}}}$ is the Fermi-Dirac distribution function,
• $E_{Fn}$ is the electron quasi-fermi level at location r,
• $E_{Fp}$ is the hole quasi-fermi level at location r,
• $T_{n}$ is the electron temperature,
• $T_{p}$ is the hole temperature,
• $f_{c}(k,r)$ is the probability that a particular conduction-band state, with wavevector k and position r, is occupied by an electron,
• $f_{v}(k,r)$ is the probability that a particular valence-band state, with wavevector k and position r, is occupied by an electron, i.e. not occupied by a hole,
• $E$ is the energy of the conduction- or valence-band state in question,
• $k_{B}$ is Boltzmann's constant.

This simplification will help us in many areas. For example, we can use the same equation for electron and hole densities used in thermal equilibrium, but substituting the Fermi levels and temperature. i.e.:

$n=n_{i}(T_{n})e^{(E_{Fn}-E_{i})/(k_{B}T_{n})}$

and

$p=p_{i}(T_{p})e^{(E_{i}-E_{Fp})/(k_{B}T_{p})}$

where

• $n$ is the density (number per unit volume) of conduction electrons,
• $p$ is the density of holes,
• $n_{i}(T_{n})$ is the density of conduction electrons that would be present in thermal equilibrium if the Fermi level were at $E_{i}$ and the temperature were $T_{n}$
• $p_{i}(T_{p})$ is the density of holes that would be present in thermal equilibrium if the Fermi level were at $E_{i}$ and the temperature were $T_{p}$,
• This equation is valid in the Boltzmann approximation, i.e. assuming the electron and hole densities are not too high.

A current (due to the combined effects of drift and diffusion) will only flow if there is a variation in the Fermi or quasi Fermi level. The current density for electron flow can be shown to be proportional to the gradient in the electron quasi Fermi level as such:

$J_{n}(r)=\mu _{n}n\nabla _{r}(E_{Fn})$, where $\mu$ is the electron mobility, and $\nabla _{r}$ is the gradient with respect to distance r.

Similarly, for holes, we have

$J_{p}(r)=\mu _{p}p\nabla _{r}(E_{Fp}).$

Notes

Nelson, Jenny. The Physics of Solar Cells. Imperial College Press, 2003.. The ratio of the areas of each pair of adjacent triangles is the same as that between the lengths of the parallel sides.

Let the trapezoid have vertices A, B, C, and D in sequence and have parallel sides AB and DC. Let E be the intersection of the diagonals, and let F be on side DA and G be on side BC such that FEG is parallel to AB and CD. Then FG is the harmonic mean of AB and DC:

${\frac {1}{FG}}={\frac {1}{2}}\left({\frac {1}{AB}}+{\frac {1}{DC}}\right).$

The line that goes through both the intersection point of the extended nonparallel sides and the intersection point of the diagonals, bisects each base.

Other properties

The center of area (center of mass for a uniform lamina) lies along the line joining the midpoints of the parallel sides, at a perpendicular distance x from the longer side b given by

$x={\frac {h}{3}}\left({\frac {2a+b}{a+b}}\right).$

If the angle bisectors to angles A and B intersect at P, and the angle bisectors to angles C and D intersect at Q, then

$PQ={\frac {|AD+BC-AB-CD|}{2}}.$

More on terminology

The term trapezium is sometimes defined in the USA as a quadrilateral with no parallel sides, though this shape is more usually called an irregular quadrilateral. The term trapezoid was once defined as a quadrilateral without any parallel sides in Britain and elsewhere, but this does not reflect current usage. (The Oxford English Dictionary says "Often called by English writers in the 19th century".)

According to the Oxford English Dictionary, the sense of a figure with no sides parallel is the meaning for which Proclus introduced the term "trapezoid". This is retained in the French trapézoïde, German Trapezoid, and in other languages. A trapezium in Proclus' sense is a quadrilateral having one pair of its opposite sides parallel. This was the specific sense in England in 17th and 18th centuries, and again the prevalent one in recent use. A trapezium as any quadrilateral more general than a parallelogram is the sense of the term in Euclid. The sense of a trapezium as an irregular quadrilateral having no sides parallel was sometimes used in England from c. 1800 to c. 1875, but is now obsolete. This sense is the one that is sometimes quoted in the US, but in practice quadrilateral is used rather than trapezium.

Architecture

The Temple of Dendur in the Metropolitan Museum of Art, New York

In architecture the word is used to refer to symmetrical doors, windows, and buildings built wider at the base, tapering towards the top, in Egyptian style. If these have straight sides and sharp angular corners, their shapes are usually isosceles trapezoids. This was the standard style for the doors and windows of the Incas.

See also

• Polite number, also known as a trapezoidal number
• Trapezoidal rule

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

External links

• Trapezoid definition   Area of a trapezoid   Median of a trapezoid With interactive animations
• Trapezoid (North America) at elsy.at: Animated course (construction, circumference, area)
• [1] on Numerical Methods for Stem Undergraduate
• Autar Kaw and E. Eric Kalu, Numerical Methods with Applications, (2008) [2]