The Sum of an Interior and Exterior Angle of a Polygon Equals 180 Degrees
(KS2, Year 6)
The sum of an interior and exterior angle in a polygon is 180°:

sum interior exterior angle 180 degrees Don't forget: This interior angle and exterior angle must be at the same vertex (corner) of the polygon for this to be the case.

How to Find a Missing Angle Using the Sum of the Interior and Exterior Angle in a Polygon Equalling 180°

Because an interior angle and an exterior angle at a vertex sum to 180°, we can find:
  • a missing interior angle when the exterior angle is known.
  • a missing exterior angle when the interior angle is known.

Finding a Missing Interior Angle

Finding a missing interior angle is easy.

Question

What is the interior angle θ of the polygon below?
sum interior exterior angle 180 degrees example 1

Step-by-Step:

1

The interior and exterior angle add up to 180°.
θ + 120° = 180°

2

Find θ by subtracting the exterior angle (120°) from 180°
θ = 180° − 120° = 60°

Answer:

The interior angle is 60°.

Finding a Missing Exterior Angle

Finding a missing exterior angle is easy.

Question

What is the exterior angle Φ of the polygon below?
sum interior exterior angle 180 degrees example 2

Step-by-Step:

1

The interior and exterior angle add up to 180°.
100° + Φ = 180°

2

Find Φ by subtracting the interior angle (100°) from 180°
Φ = 180° − 100° = 80°

Answer:

The exterior angle is 80°.

Lesson Slides

The slider below shows another real example of the sum of an interior and exterior angle in a polygon equalling 180°.

Beware

Interior and Exterior Angles at the Same Vertex Add up to 180°

Only interior and exterior angles at the same vertex (corner) of the polygon add up to 180°. If they are at a different vertex, there is not necessarily any relationship between them.

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This page was written by Stephen Clarke.