Sum of Angles of n-sided Polygon

The sum of the angles of an n-sided polygon is (n - 2) × 180°; where n is equal to the number of sides of the polygon.

Example: A pentagon is a 5-sided polygon, and the sum of its angles is equal to (5-2)×180°. 5 - 2 = 3 and 3 × 180° = 540°, which means that the sum of the angles of a pentagon is equal to 540°.

In harmony, a standard pentagon is a regular pentagon. The beauty about a regular pentagon is that all its sides are of equal magnitude. After all, the harmonious world is about equality and balance.

Since there are five angles in a pentagon, each angle of a regular pentagon is equal to the sum of the angles of a regular pentagon divided by the number of sides of the pentagon.

Let the sum of the angles of the regular pentagon be S, the number of sides n and each angle A.

Then,

A = S/n

Since S = 540°,
n = 5 and
A = 540°/5
   = 108°

Question 1:
what is the sum of the angles of a regular hexagon?

Solution:
A hexagon is a 6-sided polygon, which means that n = 6.

Sum of angles of a polygon = (n - 2) × 180°.

Sum of angles of a regular hexagon = (6 - 2) × 180°

= 4 × 180°
= 720°


Question 2:
What is the magnitude of each angle of a regular octagon?

Solution:
An octagon is an 8-sided polygon, which means that n = 8.

S (Sum of the angles of a polygon) = (n - 2) × 180°.

Substituting 8 for n,
S = (n - 2) × 180°
   = 1080°

A (an angle) = S/n
                       = 1080°/8
                       = 135°

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