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
.JPG)
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
