4.1.5 ANGLE PROPERTIES OF POLYGONS
A polygon is a closed shape with at least three sides. Polygons can be regular or irregular. Regular polygons have all their sides equal, all interior angles equal and all exterior angles equal.
Regular polygon Irregular polygon
A polygon is named according the number of sides it has.
| Number of sides | Polygon name |
|---|---|
| 3 | Triangle |
| 4 | Quadrilateral |
| 5 | Pentagon |
| 6 | Hexagon |
INTERIOR ANGLES OF POLYGONS
Interior angles are angles formed inside the polygons.
To get the sum of interior angles in a particular polygon, we divide the polygon into triangles from the vertices of the polygon as shown.
The sum of interior angles = Number of triangles × 180°.
| Polygon | No. of triangles | Sum of interior angles |
|---|---|---|
| Quadrilateral | 2 | 2 × 180° = 360° |
| Pentagon | 3 | 3 × 180° = 540° |
| Hexagon | 4 | 4 × 180° = 720° |
The general formula of getting the number of triangles in a polygon of side n is (n − 2).
The sum of interior angles of a regular polygon is given by:
Sum of interior angles = 180(n − 2)°
where n is the number of sides of the polygon.
For a regular polygon, each interior angle is obtained by dividing the sum of angles with the number of sides.
For example, a regular pentagon:
Each interior angle = 540° ÷ 5 = 108°
Example 4.9
The interior angles of a pentagon are x, 2x, 2x, x, and 3x. Calculate the value of x.
Solution.
The sum of interior angles is 180 (n − 2)
The sum of interior angles is 180 (5 − 2) = 180 × 3 = 540°
x + 2x + 2x + x + 3x = 540
9x = 540
x = 60°
EXTERIOR ANGLES OF A POLYGON
Exterior angles are formed outside the polygon when the sides of the polygon are extended.
The sum of exterior angles for any polygon is 360°.
Example 4.10
What is the size of each exterior angle in a regular hexagon?
Solution.
Since the hexagon is regular, all exterior angles are equal.
Each exterior angle = 360° ÷ 6 = 60°
Exercise
1. Calculate the value of x in each of the following.
