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§ Geometry

Polygon Properties

CCSS.5.GCCSS.7.G3 min read

Polygon properties form the backbone of KS3 and GCSE geometry, yet many pupils struggle to connect the abstract formulas with visual shapes. Teaching the relationship between sides, interior angles, and exterior angles requires systematic practice across different polygon types.

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§ 01

Why it matters

Understanding polygon properties appears throughout GCSE mathematics, from calculating angles in tessellations to solving complex geometric proofs. Architects use these principles when designing buildings—the 135° interior angles of a regular octagon create striking window designs, whilst engineers rely on hexagonal patterns for maximum strength using minimal material. In Year 8, pupils apply the (n-2)×180° formula to find that a decagon's interior angles sum to 1,440°, building foundation skills for advanced trigonometry. The exterior angle property (always summing to 360°) helps pupils understand why only certain regular polygons tessellate the plane, connecting abstract maths to real-world pattern design in everything from football manufacture to honeycomb structures.

§ 02

How to solve polygon properties

Polygon Properties

  • Sum of interior angles = (n − 2) × 180°.
  • Each interior angle of a regular n-gon = (n − 2) × 180° ÷ n.
  • Exterior angles always sum to 360°.
  • Each exterior angle of a regular n-gon = 360° ÷ n.

Example: Hexagon (n=6): sum = 4 × 180° = 720°, each = 120°.

§ 03

Worked examples

Beginner§ 01

How many sides does a pentagon have?

Answer: 5

  1. Recall the definition of a pentagon → 5 — A pentagon has 5 sides.
Easy§ 02

What is the name of a 7-sided polygon?

Answer: heptagon

  1. Match the number of sides to the polygon name → heptagon — A polygon with 7 sides is called a heptagon.
Medium§ 03

Find the interior angle of a regular decagon.

Answer: 144°

  1. Use formula: (n - 2) × 180 / n → (10 - 2) × 180 / 10 = 8 × 180 / 10 = 144° — Each interior angle of a regular decagon = (n-2)×180/n = 144°.
§ 04

Common mistakes

  • Pupils often confuse polygon names, writing 'hexagon' for a 7-sided shape instead of 'heptagon', particularly with polygons having 6-12 sides.
  • When calculating interior angles, students frequently forget the (n-2) part of the formula, writing 9×180°÷9 = 180° for a nonagon instead of (9-2)×180°÷9 = 140°.
  • Many pupils think exterior angles sum to 180° like a straight line, calculating 180°÷5 = 36° for a pentagon instead of the correct 360°÷5 = 72°.
  • Students often mix up interior and exterior angle formulas, using 360°÷n when asked for interior angles, giving 45° instead of 135° for a regular octagon.
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§ 05

Frequently asked questions

Why do exterior angles always sum to 360°?
Imagine walking around any polygon and returning to your starting position—you've made one complete turn of 360°. Each exterior angle represents how much you turn at each corner, so they must sum to 360° regardless of the polygon's shape or number of sides.
What's the difference between regular and irregular polygons?
Regular polygons have all sides equal and all angles equal (like a square or regular pentagon), whilst irregular polygons have varying side lengths or angles. The angle formulas only work directly for regular polygons—irregular polygons require individual angle calculations.
How do I remember polygon names with 6-12 sides?
Use prefixes: hexagon (6), heptagon (7), octagon (8), nonagon (9), decagon (10), hendecagon (11), dodecagon (12). Hexagon relates to 'hex' meaning 6, octagon to 'octo' meaning 8, and decagon to 'deca' meaning 10.
Why is the interior angle formula (n-2)×180°?
Any polygon can be divided into triangles from one vertex. A triangle has 3 sides and creates 1 triangle (3-2=1). A square creates 2 triangles (4-2=2). Since each triangle contains 180°, multiply by the number of triangles formed.
When do pupils learn polygon properties in the curriculum?
Basic polygon identification starts in KS2, with systematic angle calculations introduced in Year 8. GCSE Foundation requires calculating interior and exterior angles of regular polygons, whilst Higher tier extends to using these properties in complex geometric proofs and tessellation problems.
§ 06

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