Types of polygon
Polygons can be regular or irregular. If the angles are all equal, and all the sides are equal in length, it is a regular polygon.
Interior angles of polygons
To find the sum of interior angles in a polygon, divide the polygon into triangles.
The sum of interior angles in a triangle is 180°. To find the sum of interior angles of a polygon, multiply the number of triangles in the polygon by 180°.
Example
Calculate the sum of interior angles in a pentagon.
A pentagon contains 3 triangles. The sum of the interior angles is:
\({180}~\times~{3}~=~540^\circ\)
The number of triangles in each polygon is two less than the number of sides.
The formula for calculating the sum of interior angles is:
\(({n}~-~{2})~\times~180^\circ\) (where \({n}\) is the number of sides)
Question
Calculate the sum of interior angles in an octagon.
Using \(({n}~-~{2})~\times~180^\circ\) where \({n}\) is the number of sides:
\(({8}~-~{2}) \times {180}~=~1,080^\circ\)
Calculating the interior angles of regular polygons
All the interior angles in a regular polygon are equal. The formula for calculating the size of an interior angle is:
\(\text{interior~angle~of~a~polygon}\) \(\text~=~\text{sum~of~interior~angles} \div \text{number~of~sides}\)
Question
Calculate the size of the interior angle of a regular A polygon with six sides..
The sum of interior angles is \(({6}~-~{2})~\times~{180}~=~720^\circ\)
One interior angle is \({720}~\div~{6}~=~120^\circ\)
Exterior angles of polygons
If the side of a polygon is extended, the angle formed outside the polygon is the exterior angle.
The sum of the exterior angles of a polygon is 360°.
Calculating the exterior angles of regular polygons
The formula for calculating the size of an exterior angle of a regular polygon is:
\({exterior~angle~of~a~polygon}~=~{360}~\div~{number~of~sides} \)
Remember the interior and exterior angle add up to 180°.
Question
Calculate the size of the exterior and interior angle in a regular A polygon with five sides..
Method one
- The sum of exterior angles is 360°
- The exterior angle is 360 ÷ 5 = 72°
- The interior and exterior angles add up to 180°
- The interior angle is 180 - 72 = 108°
Method two
- The sum of interior angles is (5 - 2) × 180 = 540°
- The interior angle is 540 ÷ 5 = 108°
- The interior and exterior angles add up to 180°
- The exterior angle is 180 - 108 = 72°
Question
Calculate the exterior angles of the irregular pentagon below:
Using the fact that interior plus exterior angles at a point add to 180°.
Working clockwise from the top we have:
- 180 – 78 = 102°
- 180 – 99 = 81°
- 180 – 117 = 63°
- 180 – 90 = 90°
- 180 – 156 = 24°
We can check our answer by ensuring that the exterior angles add up to 360°
102 + 81 + 63 + 90 + 24 = 360°
Remember:
- The sum of interior angles in a triangle is 180°. To find the sum of interior angles of a polygon, multiply the number of triangles in the polygon by 180°.
- The formula for calculating the sum of interior angles is \(({n}~-~{2})~\times~{180^\circ}\) where \({n}\) is the number of sides.
- All the interior angles in a regular polygon are equal. The formula for calculating the size of an interior angle is: interior angle of a polygon = sum of interior angles \(\div\) number of sides.
- The sum of exterior angles of a polygon is 360°.
- The formula for calculating the size of an exterior angle is: exterior angle of a polygon = 360 \(\div\) number of sides.
