United States congressional apportionment

Angle AOB forms a central angle

A central angle is an angle whose apex (vertex) is the center O of a circle and whose legs (sides) are radii intersecting the circle in two distinct points A and B thereby subtending an arc between those two points whose angle is (by definition) equal to that of the central angle itself. It is also known as the arc segment's angular distance.

When defining or drawing a central angle, in addition to specifying the points A and B, one must specify and/or denote whether the angle being defined is the convex angle (<180°) or the reflex angle (>180°).

• The size of a central angle Θ is: 0°<Θ<360° оr 0<Θ<2π (radians)

Formulas

• If the intersection points A and B of the legs of the angle with the circle form a diameter, then Θ=180° is a straight angle. (In radians, Θ=π.)

Let L be the minor arc of the circle between points A and B, and let R be the radius of the circle.

Central angle. Convex. Includes minor arc L

• If the central angle Θ includes L, then

$0^{\circ }<\Theta <180^{\circ }\,,\,\,\Theta =\left({\frac {180L}{\pi R}}\right)^{\circ }={\frac {L}{R}}$

Proof (for degrees): The circumference of a circle with radius R is: 2πR, and the minor arc L is the (^Θ/_360°) proportional part of the whole circumference (see arc). So:

$L={\frac {\Theta }{360^{\circ }}}\cdot 2\pi R\,\Rightarrow \,\Theta =\left({\frac {180L}{\pi R}}\right)^{\circ }$

Central angle. Reflex. Does not include L

Proof (for radians): The circumference of a circle with radius R is: 2πR, and the minor arc L is the (^Θ/_2π) proportional part of the whole circumference (see arc). So:

$L={\frac {\Theta }{2\pi }}\cdot 2\pi R\,\Rightarrow \,\Theta ={\frac {L}{R}}$

• If the central angle Θ does not include the minor arc L, then the Θ is a reflex angle and:

$180^{\circ }<\Theta <360^{\circ }\,,\,\,\Theta =\left(360-{\frac {180L}{\pi R}}\right)^{\circ }=2\pi -{\frac {L}{R}}$

See also

• Inscribed angle
• Great-circle navigation

References

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External links

• Template:Cite web interactive
• Template:Cite web interactive
• Inscribed and Central Angles in a Circle