Angles in Polygons –

Angles in polygons is a core GCSE Maths topic tested at both Foundation and Higher tiers on AQA, Edexcel, and OCR papers. You need to know how to find the sum of interior angles in any polygon, calculate individual interior and exterior angles in regular polygons, and work backwards from an angle to determine the number of sides. This guide gives you every formula, walks through worked examples at both tiers, and provides practice questions to build exam confidence. For a broader review of angle rules, see our angles and parallel lines guide.

What Are Interior and Exterior Angles?

A polygon is any closed 2D shape with straight sides. Triangles, quadrilaterals, pentagons, hexagons, and so on are all polygons.

• An interior angle is the angle inside the polygon at each vertex.
• An exterior angle is the angle between one side of the polygon and the extension of the adjacent side. It is the "turn" you would make if you walked along the perimeter.

At every vertex: interior angle + exterior angle = 180° (they form a straight line).

Key Formulas

A regular polygon has all sides equal and all angles equal. For irregular polygons, you can only find the sum of the angles — individual angles will differ.

Common Polygon Names

Step-by-Step Method

Finding the Sum of Interior Angles

1. Count the number of sides (n).
2. Substitute into the formula: (n − 2) × 180°.

Finding One Interior Angle of a Regular Polygon

1. Find the sum of interior angles using (n − 2) × 180°.
2. Divide by n (since all angles are equal in a regular polygon).

Finding the Number of Sides from an Exterior Angle

1. Use the fact that each exterior angle of a regular polygon = 360° / n.
2. Rearrange: n = 360° / exterior angle.

Finding a Missing Angle in an Irregular Polygon

1. Calculate the sum of interior angles using (n − 2) × 180°.
2. Add up all the known angles.
3. Subtract the total of known angles from the sum.

Worked Example 1 — Foundation Level

Question: Find the sum of the interior angles of a heptagon (7-sided polygon). One angle is missing and the other six angles are 130°, 125°, 140°, 115°, 128°, and 135°. Find the missing angle.

Working:

Step 1 — Sum of interior angles = (7 − 2) × 180° = 5 × 180° = 900°.

Step 2 — Add the known angles: 130 + 125 + 140 + 115 + 128 + 135 = 773°

Step 3 — Missing angle = 900° − 773° = 127°.

Answer: The missing angle is 127°.

Worked Example 2 — Higher Level

Question: The interior angle of a regular polygon is 156°. How many sides does the polygon have?

Working:

Step 1 — Find the exterior angle: Exterior angle = 180° − 156° = 24°

Step 2 — Use the exterior angle formula: n = 360° / 24° = 15

Answer: The polygon has 15 sides (a regular pentadecagon).

Common Mistakes

• Using the wrong formula. Some students use 180n instead of (n − 2) × 180°. Always subtract 2 from the number of sides first.
• Confusing interior and exterior angles. Remember: they add up to 180° at each vertex. If you find one, you can always find the other.
• Dividing by n for irregular polygons. You can only divide the sum by n if the polygon is regular (all angles equal). For irregular polygons, you must be given or calculate individual angles.
• Forgetting that exterior angles sum to 360°. This is true for all convex polygons, regardless of the number of sides.
• Not checking the answer. If you find a missing angle, add all angles (including yours) to verify they sum to (n − 2) × 180°.

Exam Tips

• Start with the formula. Writing (n − 2) × 180° earns a method mark, even if you make an arithmetic error later.
• For "how many sides" questions, always find the exterior angle first, then divide 360° by it. This is the quickest approach.
• Be ready for algebra. Higher papers may give angles as expressions (e.g. 2x, 3x − 10) and ask you to form and solve an equation using the angle sum.
• Tessellation links. A regular polygon tessellates if its interior angle is a factor of 360°. Equilateral triangles (60°), squares (90°), and regular hexagons (120°) are the only ones that do.
• Draw a sketch if you are not given a diagram. Even a rough polygon helps you visualise which angles are interior and which are exterior.

Practice Questions

Question 1: Find the sum of interior angles of a nonagon (9 sides).

Question 2: Find each interior angle of a regular hexagon.

Question 3: The exterior angle of a regular polygon is 40°. How many sides does it have?

Question 4: Five angles of a hexagon are 110°, 130°, 145°, 100°, and 120°. Find the sixth angle.

Question 5: The interior angle of a regular polygon is 144°. Find the number of sides.

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Related Topics

• Angles: Basic Rules and Parallel Lines — foundational angle facts.
• Circle Theorems — angle rules inside circles (Higher only).
• Congruence and Similarity — comparing shapes with equal angles.
• Transformations — rotational symmetry of regular polygons.

Summary

Angles in polygons is tested frequently at both tiers. The two essential formulas are: the sum of interior angles = (n − 2) × 180°, and exterior angles of any convex polygon sum to 360°. For regular polygons you can find individual angles by dividing. For irregular polygons you need the individual angle values to find a missing one. Always show the formula, check your arithmetic by verifying the angle sum, and remember that interior and exterior angles at each vertex add to 180°. Mastering this topic is a reliable way to pick up marks on exam day.