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GCSE Maths / Edexcel

Interior and exterior angles

Use polygon angle facts to find interior sums, one interior angle, one exterior angle, and the number of sides of a regular polygon.

Geometry and MeasuresFoundation and HigherGrades 4 to 7Focused skill

Curriculum path: GCSE Maths > Edexcel > Geometry and Measures > Interior and exterior angles

Pearson Edexcel GCSE Maths geometry G3: derive and use the sum of angles in polygons and exterior angle facts, including regular polygons.

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Revision notes

Theory, examples, and quick checks.

Keep the method short, then practise straight away. This note is written for GCSE Maths Edexcel students who need clear working and reliable method marks.

Theory

A polygon is a flat shape with straight sides. Triangle, quadrilateral, pentagon, hexagon and octagon are common GCSE words.

Interior angles are inside the polygon. Exterior angles are made by extending a side outside the polygon.

A polygon with n sides can be split from one vertex into n - 2 triangles. That is why the interior angle sum is (n - 2) x 180 degrees.

Do not use n x 180. The two sides that meet at the starting vertex do not make extra triangles.

A regular polygon has all sides equal and all angles equal. For one interior angle, find the interior sum first, then divide by the number of sides.

The exterior angles of any polygon add to 360 degrees because walking around the shape makes one full turn.

For a regular polygon, one exterior angle is 360 divided by the number of sides.

An interior angle and the exterior angle next to it lie on a straight line, so they add to 180 degrees.

Reverse questions are common: if you know one exterior angle, divide 360 by it to find the number of sides.

Higher-style questions may hide the polygon in algebra, for example each exterior angle is x degrees or each interior angle is 150 degrees.

Key ruleInterior sum = (n - 2) x 180 degrees. Exterior angles add to 360 degrees. In a regular polygon, divide by the number of sides.

Diagram guide

3 trianglesinterior sum = (5 - 2) x 180°exterior turn
Polygon angle structureSplitting a polygon into triangles explains the interior angle formula. Exterior angles make one full turn.
exteriorexteriorone full turnall exterior angles add to 360 degrees
Exterior anglesExterior angles add to 360 degrees for every polygon. In a regular polygon, each exterior angle is equal.

Worked examples

Interior sum

Find the sum of the interior angles of a hexagon.

3 trianglesinterior sum = (5 - 2) x 180°exterior turn
Example: split into trianglesA hexagon splits into 6 - 2 = 4 triangles.
  1. A hexagon has 6 sides.
  2. Interior sum = (6 - 2) x 180.
  3. 4 x 180 = 720.

Answer: 720 degrees

Regular polygon interior angle

Find each interior angle of a regular pentagon.

3 trianglesinterior sum = (5 - 2) x 180°exterior turn
Example: regular pentagonRegular means all five interior angles are equal.
  1. Interior sum = (5 - 2) x 180 = 540.
  2. A regular pentagon has 5 equal angles.
  3. 540 / 5 = 108.

Answer: 108 degrees

Exterior angle

Find each exterior angle of a regular octagon.

exteriorexteriorone full turnall exterior angles add to 360 degrees
Example: regular exterior angleThe eight equal exterior angles share one full turn.
  1. Exterior angles add to 360 degrees.
  2. A regular octagon has 8 equal exterior angles.
  3. 360 / 8 = 45.

Answer: 45 degrees

Find the number of sides

Each exterior angle of a regular polygon is 30 degrees. Find the number of sides.

exteriorexteriorone full turnall exterior angles add to 360 degrees
Example: reverse exterior-angle questionUse the full turn of 360 degrees, then divide by one exterior angle.
  1. Exterior angles add to 360 degrees.
  2. Number of sides = 360 / 30.
  3. 360 / 30 = 12.

Answer: 12 sides

Common mistakes

  • Using n x 180 instead of (n - 2) x 180.
  • Dividing by the number of sides before finding the interior angle sum.
  • Thinking exterior angles add to 180 degrees.
  • Forgetting that regular means all angles are equal.
  • Using the exterior-angle method for a polygon that is not regular.
  • Finding the exterior angle but forgetting to subtract from 180 degrees for the interior angle.

Quick exercise

Try these before moving to the exam-style questions.

  1. Find the interior angle sum of a pentagon.
    3 trianglesinterior sum = (5 - 2) x 180°exterior turn
    Quick check: pentagonUse (n - 2) x 180 degrees.
  2. Find the interior angle sum of an octagon.
    3 trianglesinterior sum = (5 - 2) x 180°exterior turn
    Quick check: octagonAn octagon splits into 6 triangles.
  3. Find each interior angle of a regular hexagon.
    3 trianglesinterior sum = (5 - 2) x 180°exterior turn
    Quick check: regular polygonFind the total interior angle sum, then divide by 6.
  4. Find each exterior angle of a regular decagon.
    exteriorexteriorone full turnall exterior angles add to 360 degrees
    Quick check: exterior angleThe ten equal exterior angles add to 360 degrees.
  5. A regular polygon has exterior angle 30 degrees. How many sides does it have?
    exteriorexteriorone full turnall exterior angles add to 360 degrees
    Quick check: reverse questionNumber of sides = 360 divided by one exterior angle.
  6. A regular polygon has interior angle 150 degrees. Find one exterior angle.
    exteriorexteriorone full turnall exterior angles add to 360 degrees
    Quick check: interior to exteriorInterior and exterior angles on a straight line add to 180 degrees.
Exam-style questions

Practise the same skill at three levels.

These are original GCSE-style questions with mark schemes, common wrong answers, and AI marking guidance so feedback stays close to exam expectations.

Basic GCSE styleFoundationNon-calculator2 marks

Work out the sum of the interior angles of a heptagon.

3 trianglesinterior sum = (5 - 2) x 180°exterior turn
Question diagram: heptagon angle sumA heptagon splits into 7 - 2 = 5 triangles.
polygon anglesinterior angle sumfoundation geometry
Standard exam styleFoundation and HigherNon-calculator3 marks

Find each interior angle of a regular nonagon.

3 trianglesinterior sum = (5 - 2) x 180°exterior turn
Question diagram: regular nonagonRegular means the 9 interior angles are equal.
regular polygoninterior anglemethod marks
ChallengeHigherEither4 marks

Each exterior angle of a regular polygon is 24 degrees. Work out the number of sides.

exteriorexteriorone full turnall exterior angles add to 360 degrees
Question diagram: exterior angle reverseThe equal exterior angles share the full 360-degree turn.
exterior anglesregular polygonhigher geometry