|
|
| Line 1: |
Line 1: |
| {{confused|apothegm}}
| | The author's name is Andera and she thinks it seems fairly great. Invoicing is what I do. North Carolina is where we've been residing for many years and will by no means transfer. To play lacross is one of the issues she enjoys most.<br><br>Feel free to surf to my blog post - [http://www.onbizin.co.kr/xe/?document_srl=320614 online psychic chat] |
| [[Image:Apothem of hexagon.svg|thumb|right|Apothem of a [[hexagon]]]]
| |
| | |
| The '''apothem''' of a [[regular polygon]] is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the polygon that is perpendicular to one of its sides. The word "apothem" can also refer to the length of that line segment. Regular polygons are the only polygons that have apothems. Because of this, all the apothems in a polygon will be [[congruence (geometry)|congruent]]. | |
| | |
| For a regular [[pyramid (geometry)|pyramid]], which is a pyramid whose base is a regular polygon, the apothem is the slant height of a lateral face; that is, the shortest distance from apex to base on a given face. For a truncated regular pyramid (a regular pyramid with some of its peak removed by a [[plane (geometry)|plane]] parallel to the base), the apothem is the height of a trapezoidal lateral face. [http://www.bymath.com/studyguide/geo/sec/geo15.htm]
| |
| | |
| For an equilateral triangle, the apothem is equivalent to the line segment from the midpoint of a side to any of the triangle's [[Centre (geometry)#Triangles|centers]], since an equilateral triangle's centers coincide as a consequence of the definition.
| |
| | |
| ==Properties of apothems==
| |
| | |
| The apothem ''a'' can be used to find the area of any regular n-sided polygon of side length ''s'' according to the following formula, which also states that the area is equal to the apothem multiplied by half the perimeter since ''ns'' = ''p''.
| |
| :<math>A = \frac{nsa}{2} = \frac{pa}{2}. </math>
| |
| This formula can be derived by partitioning the n-sided polygon into ''n'' [[Congruence (geometry)|congruent]] [[triangle#Types of triangle|isosceles triangles]], and then noting that the apothem is the height of each triangle, and that the area of a triangle equals half the base times the height.
| |
| | |
| An apothem of a regular polygon will always be a radius of the [[inscribe]]d circle. It is also the minimum distance between any side of the polygon and its center.
| |
| | |
| This property can also be used to easily derive the formula for the area of a circle, because as the number of sides approaches infinity, the regular polygon's area approaches the area of the inscribed circle of radius ''r'' = ''a''.
| |
| | |
| :<math>A = \frac{pa}{2} = \frac{(2\pi r)r}{2} = \pi r^2</math>
| |
| | |
| ==Finding the apothem==
| |
| | |
| The apothem of a regular polygon can be found multiple ways, of which two are described here.
| |
| | |
| The apothem ''a'' of a regular ''n''-sided polygon with side length ''s'', or [[circumradius]] ''R'', can be found using the following formula:
| |
| | |
| :<math>a=\frac{s}{2\tan(\pi/n)}=R\cos(\pi/n).</math>
| |
| | |
| The apothem can also be found by
| |
| | |
| :<math>a=\frac{1}{2}s\tan\!\left(\frac{\pi(n-2)}{2n}\right).</math>
| |
| | |
| Both formulas can still be used even if only the perimeter ''p'' and the number of sides ''n'' are known because <math>s = \frac{p}{n}.</math>
| |
| | |
| == See also ==
| |
| | |
| * [[Regular polygon#Circumradius|Circumradius of a regular polygon]]
| |
| | |
| == External links ==
| |
| | |
| * [http://www.mathopenref.com/polygonapothem.html Apothem of a regular polygon] With interactive animation
| |
| * [http://www.bymath.com/studyguide/geo/sec/geo15.htm Apothem of pyramid or truncated pyramid]
| |
| * [http://demonstrations.wolfram.com/SagittaApothemAndChord/ Sagitta, Apothem, and Chord] by [[Ed Pegg, Jr.]], [[The Wolfram Demonstrations Project]].
| |
| | |
| [[Category:Polygons]]
| |
The author's name is Andera and she thinks it seems fairly great. Invoicing is what I do. North Carolina is where we've been residing for many years and will by no means transfer. To play lacross is one of the issues she enjoys most.
Feel free to surf to my blog post - online psychic chat