GCSE — mathsFoundation2 min read

Angles in Polygons

Calculate interior and exterior angles of regular and irregular polygons for GCSE Maths.

Tutor AI Team
Tutor AI Team

Polygon angle rules are fundamental GCSE geometry. You need to calculate interior and exterior angles for both regular and irregular polygons.

Core Formulas

Sum of Interior Angles

Sum=(n−2)×180°\text{Sum} = (n - 2) \times 180°

Triangle: (3−2)×180=180°(3-2) \times 180 = 180°. Quadrilateral: 360°360°. Pentagon: 540°540°.

Interior Angle of a Regular Polygon

Each angle=(n−2)×180n\text{Each angle} = \frac{(n-2) \times 180}{n}

Exterior Angles

  • Sum of exterior angles = 360°360° (always, for any convex polygon).
  • Each exterior angle (regular) = 360n\frac{360}{n}.
  • Interior + Exterior = 180°180°.

Worked Example: Example 1

Problem

Sum of interior angles of a heptagon (7 sides): (7−2)×180=900°(7-2) \times 180 = 900°.

Solution

Worked Example: Example 2

Problem

Each interior angle of a regular polygon is 156°156°. Find nn.

Exterior = 180−156=24°180 - 156 = 24°. n=36024=15n = \frac{360}{24} = 15 sides.

Solution

Worked Example: Example 3

Problem

A pentagon has angles 100°,120°,110°,95°100°, 120°, 110°, 95°, and xx. Find xx.

Sum = 540°540°. x=540−425=115°x = 540 - 425 = 115°.

Solution

Practice Problems

    1. Sum of interior angles of a nonagon (9 sides).
    1. Each exterior angle of a regular polygon is 30°30°. How many sides?
    1. Find the missing angle in a hexagon: 130°,110°,125°,140°,100°130°, 110°, 125°, 140°, 100°, and xx.

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Key Takeaways

  • ✓

    Interior sum = (n−2)×180°(n-2) \times 180°.

  • ✓

    Exterior angles always sum to 360°360°.

  • ✓

    Interior + Exterior = 180°180°.

  • ✓

    Find nn from exterior angle: n=360ext anglen = \frac{360}{\text{ext angle}}.

Tags:
#gcse#maths#geometry#angles#polygons

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