# Angles in Regular Polygon

Previously, we’ve learned solving problems about complementary and supplementary angles. Now, another basic concept and technique will be discussed about the angles in a regular polygon.

Definition of terms:

Central Angle – Angle in green, central angle is an angle made from the center of a polygon to any of the two adjacent vertices.

Interior Angle – Angle in blue, angle formed inside a polygon by connecting two adjacent sides.

Exterior Angle – Angle in brown,an angle formed by connecting subtended line of one side and adjacent side. Also defined mathematically as the supplement of interior angle.

Formula Derivations:

The formula is derived primarily from the sum of interior angles. The sum of interior angle is found by counting the number of triangles formed connecting the center of the regular polygon and its vertices. Multiplied by the sum of interior angle of triangle which is 180° and subtracting 360° which is the interior angle included included in the calculation.

The number of triangles formed in through the center is equivalent to the number sides the polygons has. Granting we have a polygon with n number of sides, we can solve the sum of interior angle (Sn ) by the following formula.

$S_n=n\cdot 180-360$

Factoring out 180° we have,

$\boxed{S_n=180(n-2)}$

Solving for the measure of interior angle:

Dividing the sum of interior angle by the number of sides of a polygon will give us the measure of interior angle (IA)

$IA=\displaystyle\frac{S_n}{n}$

$\boxed{IA=\displaystyle\frac{180(n-2)}{n}}$

Solving for the measure of Exterior angle (EA):

Using the mathematical definition of exterior angle, we have

$EA+IA=180$

But   $IA=\displaystyle\frac{180(n-2)}{n}$

$EA+\displaystyle\frac{180(n-2)}{n}=180$

$EA=180-\displaystyle\frac{180(n-2)}{n}$

$EA=180-\displaystyle\frac{180n-360}{n}$

$EA=180-\big(\displaystyle\frac{180n}{n}-\displaystyle\frac{360}{n}\big)$

$EA=180-\big(180n-\displaystyle\frac{360}{n}\big)$

$\boxed{EA=\displaystyle\frac{360}{n}}$

That’s basically the three common formula used for solving the angle of a regular polygon. Let us now go to problem solving which is the exciting part.

Worked Problem 1:

Find the sum of interior angle of a regular nonagon.

Solution:

Regular nonagon is a polygon with 9 sides. Hence n=9. Using the sum of interior angle formula we have,

$S_n=180(n-2)$

$S_9=180(9-2)$

$S_9=1260^\circ$

Worked Problem 2:

The measure of interior angle of a regular polygon is twice the measure of its exterior angle. Name the polygon.

Solution:

Polygons are named according to their number of sides, thus, we need to solve for the number of sides which is n.

Since interior and exterior angles are supplementary, we have,

$EA+IA=180$

But $IA=2EA$

$EA+2EA=180$

$3EA=180$

$EA=60^\circ$

Recall the third formula about the measure of exterior angle in terms of number of sides.

$EA=\displaystyle\frac{360}{n}$

This can be written in terms of n as,

$n=\displaystyle\frac{360}{EA}$

By substitution, we have

$n=\displaystyle\frac{360}{60}$

$n=6$

The name of a six sided polygon is hexagon.

Worked Problem 3:

Each of the interior angle of a regular polygon measures 162°, how many sides the polygon has?

Solution:

Using the second formula, we

$IA=\displaystyle\frac{180(n-2)}{n}$

$162=\displaystyle\frac{180(n-2)}{n}$

$162n=180n-360$

$180n-162n=360$

$18n=360$

$n=20$