Sheet № 152 · Foundation + Higher · AQA · Edexcel · OCR
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 si
§Key definitions
Question:
Find the sum of the interior angles of a hexagon.
Answer:
The sum of the interior angles is 720°.
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?
§Formulas to memorise
Sum of interior angles = (n − 2) × 180°, where n is the number of sides
Each interior angle of a regular polygon = ((n − 2) × 180°) ÷ n
Sum of exterior angles of any convex polygon = 360°
Each exterior angle of a regular polygon = 360° ÷ n
Interior angle + exterior angle = 180°
Count the number of sides — (n) of the polygon.
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.
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
Find the sum of the interior angles of a hexagon.
Working:
⚠ 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.