Interior Angles of Polygons
GCSE Mathematics Year 10 Understanding angle relationships in polygons
Learning Objectives
Understand what interior angles are in polygons Learn the formula for calculating the sum of interior angles Apply the formula to solve problems Identify and avoid common mistakes Solve exam-style questions confidently
What are Interior Angles?
Interior angles are angles inside a polygon They are formed between adjacent sides Each vertex of a polygon has one interior angle The sum of all interior angles follows a pattern
Interior Angle Formula
Why Does This Formula Work?
Any polygon can be divided into triangles Each triangle has angles totaling 180° A polygon with n sides creates (n-2) triangles Therefore: Sum = (n-2) × 180°
Worked Example 1
Find the sum of interior angles in an octagon Step 1: Identify n (number of sides) Step 2: Apply formula (n-2) × 180° Step 3: Calculate (8-2) × 180° = 6 × 180° = 1080°
Worked Example 2
A regular pentagon - find each interior angle Step 1: Find sum of all angles: (5-2) × 180° = 540° Step 2: Divide by number of angles: 540° ÷ 5 Step 3: Each angle = 108°
Exam-Style Question 1
The sum of interior angles of a polygon is 1260°. How many sides does this polygon have? Show your working clearly.
Exam-Style Question 2
A regular polygon has interior angles of 140° each. a) How many sides does it have? b) What is the sum of all its interior angles? Show all your working.
Common Mistakes to Avoid
Forgetting to subtract 2 from the number of sides Mixing up interior and exterior angles Not showing working clearly in exams Assuming all polygons are regular Calculation errors with large numbers
Summary and Key Points
Interior angles are angles inside a polygon Formula: Sum = (n-2) × 180° For regular polygons: Each angle = Sum ÷ n Always show your working in exams Practice with different polygon types
Answers and Solutions
Question 1: The polygon has 9 sides Working: 1260° ÷ 180° = 7, so n-2 = 7, therefore n = 9 Question 2a: The polygon has 9 sides Question 2b: Sum of angles = 1260° Remember to always show your method clearly!