Sheet № 55 · Foundation + Higher · AQA · Edexcel · OCR
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 gui
§Key definitions
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.
Answer:
The missing angle is 127°.
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?
§Formulas to memorise
Sum of interior angles = (n − 2) × 180°, where n is the number of sides
Sum of exterior angles of any convex polygon = 360°
Each interior angle of a regular polygon = (n − 2) × 180° / n
Each exterior angle of a regular polygon = 360° / n
Divide by n (since all angles are equal in a regular polygon).
Use the fact that each exterior angle of a regular polygon = 360° / n.
Rearrange: n = 360° / exterior angle.
Worked example
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:
⚠ 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.