Saturday, March 21, 2015

Regular Polygons: Internal Angles and Area

Find the Total Amount of Interior Angles

Start with a rectangular polygon where each side of length s.  This referrers to polygons with n sides.  (The pictures show a regular pentagon, where n = 5).   Draw a line from each vertex (corner) to the center of the polygon.  Note that n triangles are formed.  Label each internal angle as θ.

Note that each triangle has 180⁰ in angles.  Two of the angles of each triangle have measure half of internal angles (θ/2), and the third form a central angle.  Note that the sum of all the angles formed by the n triangles are 180⁰ * n, and:

180⁰ * n  = all interior angles + all central angles

The total of all central angle is 360⁰.  Hence:

180⁰ * n = all interior angles + 360⁰

(I)  all interior angles = 180⁰ * n - 360⁰

To find the angle of each interior angle, divide (I) by n:

(II)   each interior angle = θ = 180⁰ - 360⁰/n

Area of a Regular Polygon

Take one of the n triangles.  Determine by the height h by

tan(θ/2) = h/(s/2)
(III)  h = (s/2) * tan(θ/2)

And the area of each triangle is:

area = 1/2 * base * height
area = 1/2 * s * (s/2) * tan(θ/2)
(IV)   area = 1/4 * s^2 * tan(θ/2)

Taking each of the n triangles are into account, the total area (A) of regular polygon is:

A = n * area
(V)  A = n/4 *s^2 * tan(θ/2)

Note, we can state the area of the regular polygon in a separate form.

Substitute (II) into (V):

θ = 180⁰ - 360⁰/n
A = n/4 *s^2 * tan(1/2 *(180⁰ - 360⁰/n))
A = n/4 * s^2 * tan(90⁰ - 180⁰/n)

By the trigonometric identity cot(x) = tan(90⁰ - x) (see picture above),

(VI)  A = n/4 * s^2 * cot(180⁰/n)

To summarize, for a regular polygon:

Total of the interior angles:  180⁰ * n - 360⁰
Each interior angle:  θ = 180⁰ - 360⁰/n
Area of the regular polygon: A = n/4 *s^2 * tan(θ/2) = n/4 * s^2 * cot(180⁰/n)

This blog is property of Edward Shore.  2015

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