IGCSEMathsCambridge (CIE)CoreRevision NotesGeometryAngles in Polygons & Parallel LinesAngles in Polygons

Angles in Polygons (Cambridge (CIE) IGCSE Maths: Core): Revision Note

Exam code: 0580 & 0980

Written by: Jamie Wood

Reviewed by: Dan Finlay

Updated on 27 January 2026

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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 is a 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

Interior and exterior angles in a hexagon

The interior angle and exterior angle add up to 180° at each corner

interior and exterior angles summing to 180 degrees

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:
Method 1
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: $540 ° \div 5 = 108 °$
Method 2
Use the formula $\frac{180 (n - 2)}{n}$ which calculates the interior angle directly
To find the size of an exterior angle in a regular polygon:
- either divide 360° by the number of sides ( $n$ )
  - For a pentagon: $360 ° \div 5 = 72 °$
- or use the formula $\frac{360}{n}$
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?

To find a missing angle in a polygon:
- Use the formula $180 ° \times (n - 2)$ to work out the sum of the interior angles
- Subtract the other interior angles in the polygon

How do I find the number of sides in a regular polygon?

If you are given the interior angle of a regular polygon
- set the angle equal to $\frac{180 (n - 2)}{n}$
- then solve the equation to find $n$

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.

Answer:

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

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Questions students ask about Angles in Polygons

Real questions from students revising this topic on Save My Exams

What is an exterior angle of a polygon?

An exterior angle is the angle between one side of a polygon and the extension of the side next to it — it makes a straight line, 180°, with the interior angle at that corner.

Extend one side past the corner with a dashed line, as in the page's diagrams
The angle between that extension and the next side is the exterior angle
It is not the angle that makes a full 360° turn at the corner

Key facts:

interior angle + exterior angle = 180° at every corner
The exterior angles of any polygon add up to 360°
In a regular polygon with n sides, each exterior angle = 360° ÷ n

Example — a regular pentagon:

exterior angle = 360° ÷ 5 = 72°
interior angle = 180° − 72° = 108°

Check against the page's table: a square has an exterior angle of 90° and an interior angle of 90°; a regular hexagon has 60° and 120°.

Picture walking round the edge of the shape: the exterior angle is how far you turn at each corner, and one full lap turns you through 360°.

How do you find the size of an interior angle of a polygon?

For a regular polygon, divide the sum of the interior angles, 180° × (n − 2), by the number of sides n.

Regular polygon — three routes, all giving the same answer. For a regular pentagon (n = 5):

Sum, then divide: 180° × (5 − 2) = 540°, and 540° ÷ 5 = 108°
Direct formula: 180(n − 2)/n = 180 × 3 ÷ 5 = 108°
Via the exterior angle: 360° ÷ 5 = 72°, then 180° − 72° = 108°

Irregular polygon (a missing angle)

Find the sum using 180° × (n − 2)
Subtract the interior angles you already know

Example: a pentagon with angles 100°, 110°, 120° and 95°

Sum = 540°, and 100 + 110 + 120 + 95 = 425°
Missing angle = 540° − 425° = 115°

Sums worth remembering: triangle 180°, quadrilateral 360°, pentagon 540°

Always check whether the polygon is regular before dividing — dividing the sum by n only works when all the angles are equal.

How do you calculate the number of sides of a polygon?

For a regular polygon, divide 360° by one exterior angle: n = 360° ÷ exterior angle.

Given the exterior angle (the page's example)

The exterior angle of a regular polygon is 45°:

360° ÷ n = 45°
n = 360 ÷ 45 = 8
8 sides, so it is a regular octagon

Given the interior angle

Find the exterior angle first, then divide it into 360°. For an interior angle of 144°:

exterior angle = 180° − 144° = 36°
n = 360 ÷ 36 = 10, a regular decagon

Or, as the page says, set the interior angle equal to 180(n − 2)/n and solve:

180(n − 2) = 144n
180n − 360 = 144n, so 36n = 360 and n = 10

Given the sum of the interior angles

Solve 180(n − 2) = sum. For a sum of 1080°: n − 2 = 1080 ÷ 180 = 6, so n = 8

If n does not come out as a whole number, no regular polygon has that angle.