Module 6 (M6) - Geometry and measures - Angles in polygons

Part ofMathsM6: Geometry and measures

Angles in Polygons

It may be useful to refer to M1 properties of angles; M1 properties of 2D shapes and M5 Polygons if necessary.

Blue triangle
The internal angles of a triangle always add up to 180°

The sum of the angles inside any polygon can be found by spitting the polygon into triangles.

Consider any triangle. The sum of the angles is known to be 180°

Blue triangle
The internal angles of a triangle always add up to 180°
Blue quadrilateral divided into two triangles. Angles marked
A quadrilateral divided into two triangles

A quadrilateral can be split into two triangles. Each triangle has 180°.

The sum of the angles in a quadrilateral is 2 x 180° = 360°

Irregular polygons
A pentagon divided into three triangles and a hexagon divided into 4 triangles

  • The sum of the angles in a pentagon is 3 × 180° = 540°

  • The sum of the angles in a hexagon is 4 × 180° = 720°

The number of triangles is always two less than the number of sides.

The formula for finding the sum of the interior angles in any polygon is:

Sum of interior angles = two less than the number of sides × 180°

In algebra this is

Sum of interior angles = 180(n – 2)
where n is the number of sides

In a regular polygon all the angles are equal. Each interior angle is calculated by dividing the sum of the interior angles by the number of sides.

Interior angle in a regular polygon = \(\mathbf{\frac{180(n-2)}{n}}\)

Example

  1. Calculate the sum of the angles in a regular octagon.
  2. Calculate the size of each angle.
Diagram of an octagon

Solution:

  1. Sum of interior angles = 180(n – 2)
    For an octagon n = 8
    Sum of interior angles = 180(8– 2) = 180 x 6 = 1080°
  2. Interior angle in a regular polygon = \(\frac{180(n-2)}{n} \)
    \(\frac{1080}{n} = \frac{1080}{8} = 135\textdegree\)

Question

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

A regular polygon with the equal angles coloured green.

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Exterior angles in 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°.

An irregular polygon with the external multi-coloured angles, to show they add up to 360°.

Calculating the size of each exterior angle of regular polygons

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

\(\text{exterior angle of a regular polygon}~=~{360°}~\div~\text{number of sides}\)

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

Example

A regular polygon has an exterior angle of 12°.

How many sides does it have?

Solution:

We know that the exterior angle \( = \frac{360\textdegree}{(number~of~sides)}\)

By rearranging this formula we get

\(Number~of~sides = \frac{360\textdegree}{(exterior~angle)}\) When the exterior angle is 12° \(Number~of~sides = \frac{360}{12} = 30\)

The polygon has 30 sides

Question

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

A pentagon showing that an external angle (green) and an internal angle (yellow) add to 180°

Question

Calculate the exterior angles of the irregular pentagon below:

An irregular pentagon with the interior angles marked (from left to right) 90°, 156°, 78°, 99°, 117°.

Remember:

  • The sum of the angles in a triangle is 180°. To find the sum of the interior angles of a polygon, multiply the number of triangles in the polygon by 180°.
  • The formula for calculating the sum of the interior angles in a polygon 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 in a regular polygon is: the sum of interior angles \(\div\) number of sides.
  • The sum of the exterior angles of a polygon is 360°.
  • The formula for calculating the size of an exterior angle in a regular polygon is: 360 \(\div\) number of sides.
  • If you know the exterior angle you can find the interior angle using the formula: interior angle + exterior angle = 180°
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Test yourself

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