GCSE MATHS • HIGHER TIER

Polygons Made Simple

Learn polygon names, regular and irregular polygons, interior and exterior angle rules, and how to find missing angles and numbers of sides.

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Polygons in Four Moves

1

Identify

Identify the polygon and count its number of sides.

2

Choose

Decide whether the question involves interior angles, exterior angles or a regular polygon.

3

Calculate

Apply the correct formula or angle fact.

4

Check

Check the polygon name, number of sides and angle total.

What You’ll Learn

Name Polygons

Recognise common polygons from their number of sides.

Regular vs Irregular

Understand the difference between regular and irregular polygons.

Interior Angles

Calculate the sum of interior angles using (n − 2) × 180°.

Exterior Angles

Use the fact that exterior angles total 360°.

Find Number of Sides

Use a regular exterior angle to calculate the number of sides.

Applications

Real-Life Applications

Polygons are used in architecture, engineering, tiling, graphic design, road signs and computer modelling. Many everyday objects are based on polygon shapes.

Worked Proof: Polygons

Regular and Irregular Polygons

A regular polygon has all sides equal and all angles equal. An irregular polygon does not have equal sides and angles.

Why Are Polygons Important?

Polygons appear throughout GCSE Maths in topics such as angle calculations, tessellations, symmetry, area and perimeter. They also appear in architecture, engineering and design.

More Worked Examples

1

Interior Angles of Polygons

The sum of interior angles can be found using:

(Number of sides − 2) × 180°
Find the sum of the interior angles of a pentagon.
Step 1: Number of sides = 5
(5 − 2) × 180
3 × 180 = 540
Answer: 540°

2

Interior Angles of Polygons

The sum of interior angles can be found using:

(Number of sides − 2) × 180°
Find the sum of the interior angles of a hexagon.
(6 − 2) × 180 = 720
Answer: 720°

3

Finding One Interior Angle of a Regular Polygon

If a polygon is regular, divide the total interior angle sum by the number of sides.

A regular pentagon has an interior angle sum of 540°.
540 ÷ 5 = 108°
Answer: Each interior angle = 108°

4

Finding One Interior Angle of a Regular Polygon

If a polygon is regular, divide the total interior angle sum by the number of sides.

Exterior angles of any polygon total 360°
A regular octagon has 8 equal exterior angles.
360 ÷ 8 = 45°
Answer: 12 sides
  • Common GCSE Mistakes
  • Interior formula: Do not forget to subtract 2 from the number of sides.
  • Interior vs exterior: Check which angle type the question asks for.
  • Exterior total: Remember exterior angles always total 360°.
  • Regular angle calculation: Divide by the number of sides carefully.
  • Polygon names: Match each name to the correct number of sides.
  • Exam Tips
  • Write the interior formula first: (n − 2) × 180°.
  • For a regular interior angle: Divide the interior sum by n.
  • For a regular exterior angle: Use 360° ÷ n.
  • To find sides: Use 360° ÷ exterior angle.
  • Check the polygon name: Confirm the number of sides before calculating.

Skills Used in Polygons

Polygon Names

Interior Angles

Exterior Angles

Regular Polygons

Angle Calculations

Symmetry

Area and Perimeter

Geometry Problem Solving

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Step-by-step walkthroughs of Polygons with clear methods and exam tips.

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Frequently Asked Questions

A polygon is a closed 2D shape made entirely from straight lines.

A regular polygon has all sides equal and all angles equal.

Use (number of sides − 2) × 180°.

The exterior angles always add up to 360°.

Find the total interior angle sum and divide it by the number of sides.

For a regular polygon, divide 360° by the exterior angle.