Interior and Exterior Angles of Polygons –

Interior and exterior angles of polygons are tested on every GCSE Maths exam board. You need to know the formulas for the sum of interior angles, the size of each angle in a regular polygon, and the relationship between interior and exterior angles. This guide covers everything from Foundation to Higher, including finding the number of sides from an angle.

What Are Interior and Exterior Angles of Polygons?

An interior angle is the angle inside a polygon at each vertex. An exterior angle is formed between one side of the polygon and the extension of the adjacent side. At each vertex, the interior and exterior angles sit on a straight line, so they always add up to 180°.

Key Formulas

Step-by-Step Method

1. Count the number of sides (n) of the polygon.
2. Choose the correct formula. Use (n − 2) × 180° for the sum of interior angles, or 360° ÷ n for each exterior angle of a regular polygon.
3. Substitute and calculate.
4. If finding the number of sides, rearrange the formula. For example, if each exterior angle is given, n = 360° ÷ exterior angle.

Worked Example 1 — Foundation Level

Question: Find the sum of the interior angles of a hexagon.

Working:

A hexagon has 6 sides, so n = 6.

Sum of interior angles = (6 − 2) × 180° = 4 × 180° = 720°

Answer: The sum of the interior angles is 720°.

Worked Example 2 — Higher Level

Question: Each exterior angle of a regular polygon is 24°. How many sides does the polygon have? Find each interior angle.

Working:

Number of sides = 360° ÷ 24° = 15

Each interior angle = 180° − 24° = 156°

Answer: The polygon has 15 sides and each interior angle is 156°.

Worked Example 3 — Exam Style

Question: Four angles of a pentagon are 108°, 92°, 130°, and 115°. Find the fifth angle.

Working:

Sum of interior angles of a pentagon = (5 − 2) × 180° = 540°

Fifth angle = 540° − 108° − 92° − 130° − 115° = 540° − 445° = 95°

Answer: The fifth angle is 95°.

Common Mistakes

• Using 180° × n instead of (n − 2) × 180°. The formula subtracts 2 from the number of sides before multiplying. Forgetting this gives an answer that is 360° too large.
• Confusing interior and exterior angles. Remember that they add up to 180° at each vertex. The exterior angle formula uses 360° ÷ n, not the interior angle formula.
• Assuming all polygons are regular. The formulas for individual angles only apply to regular polygons (all sides and angles equal). For irregular polygons, you can only use the sum formula.
• Giving the exterior angle when the question asks for interior (or vice versa). Read the question twice and underline which angle type is requested.

Exam Tips

• If a question gives you an interior angle of a regular polygon, first find the exterior angle by subtracting from 180°, then divide 360° by the exterior angle to get the number of sides.
• Draw a quick sketch and mark the angles to help visualise the problem.
• Learn the common polygon names: pentagon (5), hexagon (6), heptagon (7), octagon (8), nonagon (9), decagon (10).
• For irregular polygons, find the sum of interior angles first, then subtract the known angles.
• Remember that the exterior angles of any convex polygon always sum to 360° regardless of the number of sides.
• Questions worth 3 or more marks often involve setting up an equation — for example, when one angle is given as an algebraic expression. Always show your equation and working clearly.

Practice Questions

Q1 (Foundation): Find the sum of the interior angles of an octagon (8 sides).

Q2 (Foundation): Find each exterior angle of a regular decagon (10 sides).

Q3 (Higher): The interior angle of a regular polygon is 162°. How many sides does the polygon have?

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Related Topics

• Angles on a Straight Line and at a Point
• Alternate, Corresponding and Co-interior Angles
• Angles in Polygons
• Congruent Triangles

Summary

• The sum of interior angles of any polygon with n sides is (n − 2) × 180°. Exterior angles of any convex polygon always sum to 360°. For regular polygons, each exterior angle is 360° ÷ n and each interior angle is 180° minus the exterior angle. These formulas let you find missing angles, determine the number of sides, and solve problems involving both regular and irregular polygons. Practise converting between interior and exterior angles quickly — this skill saves valuable time in multi-mark exam questions.