Angles in Polygons (Cambridge (CIE) IGCSE Maths)

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Written by: Jamie Wood

Reviewed by: Dan Finlay

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Angles in Polygons

What is a polygon?

A polygon is a 2D shape with $n$ straight sides
- A triangle is a polygon with 3 sides
- A quadrilateral polygon with 4 sides
- A pentagon is a polygon with 5 sides
In a regular polygon all the sides are the same length and all the angles are the same size
- A regular polygon with 3 sides is an equilateral triangle
- A regular polygon with 4 sides is a square

What are the interior angles and the exterior angles of a polygon?

Interior angles are the angles inside a polygon at the corners
The exterior angle at a corner is the angle needed to make a straight line with the interior angles
- It is not the angle that forms a full turn at the corner
The interior angle and exterior angle add up to 180° at each corner

interior and exterior angles summing to 180 degrees

Interior and exterior angles in a hexagon

What is the sum of the interior angles in a polygon?

To find the sum of the interior angles in a polygon of $n$ sides, use the rule
- Sum of interior angles = $180 ° \times (n - 2)$
  - This formula comes from the fact that $n$ -sided polygons can be split into $n - 2$ triangles
Remember the sums for these polygons
- The interior angles of a triangle add up to 180°
- The interior angles of a quadrilateral add up to 360°
- The interior angles of a pentagon add up to 540°

What is the sum of the exterior angles in a polygon?

The exterior angles in any polygon always sum to 360°

How do I find the size of an interior or exterior angle in a regular polygon?

To find the size of an interior angle in a regular polygon:
- Find the sum of the interior angles
  - For a pentagon: $180 ° \times (5 - 2) = 540 °$
- Divide by the number of sides ( $n$ )
  - For a pentagon: $540 ° \div 5 = 108 °$
To find the size of an exterior angle in a regular polygon:
- Divide 360° by the number of sides ( $n$ )
  - For a pentagon: $360 ° \div 5 = 72 °$
The interior angle and exterior angle add to 180°
- Subtract the exterior angle from 180° to find the interior angle
- Subtract the interior angle from 180° to find the exterior angle

Regular Polygon	Number of Sides	Sum of Interior Angles	Size of Interior Angle	Size of Exterior Angle
Equilateral Triangle	3	180°	60°	120°
Square	4	360°	90°	90°
Regular Pentagon	5	540°	108°	72°
Regular Hexagon	6	720°	120°	60°
Regular Octagon	8	1080°	135°	45°
Regular Decagon	10	1440°	144°	36°

How do I find a missing angle in a polygon?

STEP 1
Calculate the sum of the interior angles for the polygon
- Use the formula $180 ° \times (n - 2)$
STEP 2
Subtract the other interior angles in the polygon

Examiner Tips and Tricks

Make sure you identify whether you are dealing with a regular or irregular polygon before you start a question
Finding the sum of the interior angles using $180 \times (n - 2)$ can often be a good starting point for finding missing angles

Worked Example

The exterior angle of a regular polygon is 45°.

Write down the name of the polygon.

The formula for the exterior angle of a regular polygon is $Exterior Angle = \frac{360 °}{n}$

Substitute the 45 for the exterior angle

$45 ° = \frac{360 °}{n}$

Solve by rearranging

$\begin{array}{rcl} n & = & \frac{360}{45} \\ n & = & 8 \end{array}$

Write down the name of a shape with 8 sides

Regular Octagon

Last updated: 17 October 2024

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