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§ Geometry

Polygon Properties

§ Geometry

Polygon Properties

CCSS.5.GCCSS.7.G3 min read

A polygon is a closed two-dimensional shape formed by three or more straight line segments called sides. The word polygon comes from Greek, meaning 'many angles', and these shapes are classified by their number of sides: triangles have 3 sides, quadrilaterals have 4, pentagons have 5, and so forth. Regular polygons have all sides equal and all angles equal, whilst irregular polygons have varying side lengths or angles.

§ 01

Why it matters

Polygon properties appear throughout GCSE Mathematics and have extensive real-world applications. Architects use hexagonal floor tiles because regular hexagons fit together perfectly with no gaps, each interior angle measuring exactly 120°. Engineers design nuts and bolts with hexagonal shapes for optimal grip strength. The Pentagon building in Washington uses the 108° interior angles of a regular pentagon in its famous design. Construction workers rely on the 60° angles in equilateral triangles for stable roof trusses. Traffic signs utilise polygon properties: stop signs are regular octagons with 135° interior angles, whilst warning signs are equilateral triangles. Video game designers program polygon meshes using these angle calculations to render 3D objects smoothly.

§ 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 decagon have?

Answer: 10

  1. Recall the definition of a decagon → 10 — A decagon has 10 sides.
Easy§ 02

What is the name of a 3-sided polygon?

Answer: triangle

  1. Match the number of sides to the polygon name → triangle — A polygon with 3 sides is called a triangle.
Medium§ 03

Find the interior angle of a regular octagon.

Answer: 135°

  1. Use formula: (n - 2) × 180 / n → (8 - 2) × 1808 = 6 × 1808 = 135° — Each interior angle of a regular octagon = (n-2)×180/n = 135°.
§ 04

Common mistakes

  • Confusing interior and exterior angle formulas leads to calculating a regular hexagon's interior angle as 360° ÷ 6 = 60° instead of (6-2) × 180° ÷ 6 = 120°.
  • Forgetting to subtract 2 in the interior angle sum formula gives a triangle's total as 3 × 180° = 540° instead of (3-2) × 180° = 180°.
  • Mixing up polygon names by counting incorrectly results in calling a 7-sided shape a hexagon instead of a heptagon.
§ 05

Frequently asked questions

What is the difference between regular and irregular polygons?
Regular polygons have all sides equal and all angles equal, like a square or regular pentagon. Irregular polygons have sides or angles of different sizes, such as a rectangle (equal angles but unequal sides) or a scalene triangle (unequal sides and angles).
How do you calculate the interior angle of any regular polygon?
Use the formula (n-2) × 180° ÷ n, where n is the number of sides. For a regular octagon: (8-2) × 180° ÷ 8 = 6 × 180° ÷ 8 = 135°. This works because the sum of interior angles is (n-2) × 180°, divided equally among n angles.
Why do exterior angles always sum to 360°?
Imagine walking around the perimeter of any polygon and returning to your starting position. At each vertex, you turn through the exterior angle. After completing the full journey, you have turned through one complete rotation, which is always 360°, regardless of the polygon's shape or size.
What are the names of polygons with 5 to 10 sides?
Pentagon (5 sides), hexagon (6 sides), heptagon (7 sides), octagon (8 sides), nonagon (9 sides), and decagon (10 sides). These names derive from Greek and Latin roots indicating the number of sides.
Can a polygon have interior angles greater than 180°?
Yes, in concave (non-convex) polygons, some interior angles can exceed 180°. However, the formulas (n-2) × 180° for angle sum and 360° for exterior angles apply only to convex polygons, where all interior angles are less than 180°.
§ 06

See also

PerimeterVertex
§ 06

Related topics

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