**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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