area of regular polygon
Theorem 1.
Given a regular n-gon (http://planetmath.org/RegularPolygon) with apothem of length a and perimeter
(http://planetmath.org/Perimeter2) P, its area is
A=12aP. |
Proof.
Given a regular n-gon R, line segments can be drawn from its center to each of its vertices. This divides R into n congruent triangles
. The area of each of these triangles is 12as, where s is the length of one of the sides of the triangle. Also note that the perimeter of R is P=ns. Thus, the area A of R is
A=n(12as)=12a(ns)=12aP.
∎
To illustrate what is going on in the proof, a regular hexagon appears below with each line segment from its center to one of its vertices drawn in red and one of its apothems drawn in blue.
Title | area of regular polygon |
---|---|
Canonical name | AreaOfRegularPolygon |
Date of creation | 2013-03-22 17:11:06 |
Last modified on | 2013-03-22 17:11:06 |
Owner | Wkbj79 (1863) |
Last modified by | Wkbj79 (1863) |
Numerical id | 6 |
Author | Wkbj79 (1863) |
Entry type | Theorem |
Classification | msc 51-00 |