Polygons

A polygon is a 2D shape with at least three sides.

Types of polygon

Polygons can be regular or irregular. If the angles are all equal and all the sides are equal length it is a regular polygon.

Regular and irregular polygons

Interior angles of polygons

To find the sum of interior angles in a polygon divide the polygon into triangles.

Irregular pentagons

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:

\[\text{interior angle of a polygon} = \text{sum of interior angles} \div \text{number of sides}\]

Question

Calculate the size of the interior angle of a regular hexagon.

Hexagon with all internal angles highlighted

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°.

External angles produced along the sides of a pentagon equal 360 degrees

Calculating the exterior angles of regular polygons

The formula for calculating the size of an exterior angle is:

\[\text{exterior angle of a polygon} = 360 \div \text{number of sides}\]

Remember the interior and exterior angle add up to 180°.

Question

Calculate the size of the exterior and interior angle in a regular pentagon.

Pentagon with internal and external angles highlighted

Method 1

The sum of exterior angles is 360°.

The exterior angle is \(360 \div 5 = 72^\circ\).

The interior and exterior angles add up to 180°.

The interior angle is \(180 - 72 = 108^\circ\).

Method 2

The sum of interior angles is \((5 - 2) \times 180 = 540^\circ\).

The interior angle is \(540 \div 5 = 108^\circ\).

The interior and exterior angles add up to 180°.

The exterior angle is \(180 - 108 = 72^\circ\).

curriculum-key-fact
  • 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 ÷ 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 ÷ number of sides.