Polygons, Sum of Interior and Exterior Angles of a Polygon

# Polygons, Sum of Interior and Exterior Angles of a Polygon

## Polygons

A closed plane figure formed by three or more line segments is called a polygon. The word polygon is made of two words 'poly' and 'gon'. 'Poly' means 'many' and 'gon' means 'sides'. Thus, polygon means figure with many sides. Polygons are made with only line segments.

## Types of Polygons

When each interior angle of a polygon has a measure that is less than 180°, the polygon is called a convex polygon. If one or more interior angles of a polygon are reflex angles, it is called a non-convex polygon. In this section, we shall discuss the properties of convex polygons.

A polygon is named according to the number of sides it has. Here are some common polygons.

A polygon with n sides can be called an n-gon. For instance, a polygon with 17 sides is called a 17-gon.
A regular polygon has equal sides and equal angles. The figures given below show a regular pentagon and hexagon respectively.

## Sum of Interior Angles of a Polygon

All the angles inside the polygon are called its interior angles. We have learned that the angle sum of a triangle is 180°.
The sum of the interior angles of an n-sided polygon is (n – 2) × 180°.

If n = 3, then the sum of the interior angles = (3 - 2) × 180° = 180°
If n = 4, then the sum of the interior angles = (4 - 2) × 180° = 360°

Example 1: In the given figure, find the value of angle x.

Solution: x + 73° + 55° + 115° = 360° ( sum of polygon)
x = 117°

Example 2: The angle sum of the interior angles of an n-sided polygon is 3600°. Find the value of n.

Solution: Sum of the interior angles of an n-gon = (n – 2) × 180° ( sum of polygon)
(n 2) × 180° = 3600°
n – 2 = 3600/180
n – 2 = 20
n = 22

## Sum of Exterior Angles of a Polygon

A polygon has as many exterior angles as its number of sides. For example, if we produce all the sides of a quadrilateral, then four exterior angles are formed. And if we produce all the sides of a pentagon, then five exterior angles are formed.

The sum of the exterior angles of a polygon is 360°.”

Example 3: Find the measure of each exterior angle of a regular decagon.

Solution: Sum of the exterior angles of a decagon = 360° (exterior sum of polygon)
the measure of each exterior angle of a decagon = 360° ÷ 10 = 36°

Example 4: In the given figure, find the measures of angles x and y.

Solution: x + 89° = 180° (adj. s on a st. line)
x = 91°
102° + x + y + 103° = 360° (ext. sum of polygon)
102° + 91° + y + 103° = 360°
y = 64°