Angles in polygons – GCSE Maths

Foundation and Higher · Grade 4 · Non-calculator

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The interior angles of an $n$-sided polygon sum to $(n-2)\times 180^{\circ }$. A regular polygon shares that sum equally. Exterior angles of any convex polygon sum to $360^{\circ }$, and for a regular polygon each exterior angle is $360^{\circ }÷n$. Interior and exterior at a vertex sit on a straight line, so they add to $180^{\circ }$.

To find $n$ from an exterior angle, divide $360$ by that angle. Do not use the interior-sum formula to find $n$ unless you first find one exterior (or rearrange carefully). A pentagon is $5$ sides, a hexagon $6$, octagon $8$ — write the name if you know it, but $n$ is what the algebra needs.

Interior sum $=(n-2)\times 180^{\circ }$. Exterior of any convex polygon sums to $360^{\circ }$, so a regular exterior is $360÷n$. Interior of a regular polygon is $180$ minus that exterior. Working backwards from an interior angle is the usual 4-mark.

Objectives

• One exterior angle is the turn at a corner if you walk around the shape.
• At each vertex, the interior and the exterior add to 180°.
• A polygon with n sides splits into n − 2 triangles.

Key points

• Find the sum of the interior angles of any polygon using $(n-2)\times 180^{\circ }$.
• Work out one interior angle of a regular polygon.
• Use the fact that exterior angles always total $360^{\circ }$.
• Find the number of sides of a regular polygon from its exterior or interior angle.

Worked examples

Questions and answers

GCSE Maths angles in polygons questions and answers. Try each one, then open the answer. Easy questions come first.

Common mistakes

Most lost marks on this topic are one slip.

1. Dividing 180° by the number of sides to get an exterior angle.

   Exterior angles add to 360°.

2. Using (n − 1) × 180° for the interior sum.

   It is n − 2 triangles.

3. Treating an irregular polygon as if every angle were equal.

   Only a regular polygon shares one interior angle out equally.

Exam tips

• Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
• If the question says “show that”, the result is given. The marks are for the steps that reach it.
• The method is the same on AQA-style and Edexcel-style papers.
• Write the formula $(n-2)\times 180$ with your value of $n$ substituted; that line is the method mark.
• If a 'number of sides' answer is not a whole number, you have made a slip — recheck whether you used the interior or exterior angle.

Printable worksheet

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Questions students ask

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More on Angles in polygons

• Angles in polygons worked examples and method
  The method in order, key facts and the full lesson explanation
• Angles in polygons practice questions
  A second set of questions with answers, plus quick checks

AQA-style and Edexcel-style questions. Yumio is not affiliated with, endorsed by or connected to AQA, Pearson Edexcel or any other exam board.

Last updated 9 November 2026.