Difference between revisions of "Apothem"

Revision as of 17:22, 9 March 2014

The apothem of a regular polygon is the line segment drawn from the center of the polygon perpendicular to one of its edges. It is also the radius of the inscribed circle of the polygon.

Formulas

Given the number of sides, $n$ , and side length, $s$ , the length of the apothem is $\frac{s}{2\tan\left(\frac{\pi}{n}\right)}$ .
Given the number of sides, $n$ , and radius of the circumscribed circle, $R$ , the length of the apothem is $R\cos\left(\frac{\pi}{n}\right)$ .
Given the length of the apothem, $a$ , and the perimeter, $p$ , of a regular polygon, the area of the polygon is $\frac{ap}{2}$ , or $as$ , where $s$ is the semiperimeter.

$[asy] draw((0,sqrt(3))--(1,0)--(3,0)--(4,sqrt(3))--(3,2sqrt(3))--(1,2sqrt(3))--cycle); draw((1,0)--(2,sqrt(3))--(3,0),gray+linetype("1 2")); draw((2,0)--(2,sqrt(3)),gray); [/asy]$ This article is a stub. Help us out by expanding it.

@@ Line 2: / Line 2: @@
 == Formulas ==
+*Given the number of sides, <math>n</math>, and side length, <math>s</math>, the length of the apothem is <math>\frac{s}{2\tan\left(\frac{\pi}{n}\right)}</math>.
+*Given the number of sides, <math>n</math>, and radius of the [[circumcircle | circumscribed circle]], <math>R</math>, the length of the apothem is <math>R\cos\left(\frac{\pi}{n}\right)</math>.
+*Given the length of the apothem, <math>a</math>, and the [[perimeter]], <math>p</math>, of a regular polygon, the [[area]] of the polygon is <math>\frac{ap}{2}</math>, or <math>as</math>, where <math>s</math> is the [[semiperimeter]].
-Given the number of sides, <math>n</math>, and side length, <math>s</math>, the length of the apothem is <math>\frac{s}{2\tan\left(\frac{\pi}{n}\right)}</math>.
-Given the number of sides, <math>n</math>, and radius of the [[circumcircle | circumscribed circle]], <math>R</math>, the length of the apothem is <math>R\cos\left(\frac{\pi}{n}\right)</math>.
-Given the length of the apothem, <math>a</math>, and the [[perimeter]], <math>p</math>, of a regular polygon, the [[area]] of the polygon is <math>\frac{ap}{2}</math>, or <math>as</math>, where <math>s</math> is the [[semiperimeter]].
+<asy>
+draw((0,sqrt(3))--(1,0)--(3,0)--(4,sqrt(3))--(3,2sqrt(3))--(1,2sqrt(3))--cycle);
+draw((1,0)--(2,sqrt(3))--(3,0),gray+linetype("1 2"));
+draw((2,0)--(2,sqrt(3)),gray);
+</asy>
 {{stub}}
 [[Category:Definition]]
+[[Category:Geometry]]

Page

Toolbox

Search

Difference between revisions of "Apothem"

Revision as of 17:22, 9 March 2014

Formulas