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§ Geometry

Polygon Properties

§ Geometry

Polygon Properties

CCSS.5.GCCSS.7.G3 min read

A polygon is a closed figure formed by three or more straight line segments called sides, which meet at points called vertices. Each polygon is classified by its number of sides: a triangle has 3 sides, a quadrilateral has 4 sides, a pentagon has 5 sides, and so on. Regular polygons have all sides equal in length and all angles equal in measure.

§ 01

Why it matters

Polygon properties appear throughout architecture, engineering, and design. Stop signs use regular octagons because their 8 equal sides create visual balance and recognition from any angle. Soccer balls combine pentagons and hexagons — each pentagon is surrounded by 5 hexagons, creating the familiar spherical shape through 32 total faces. In computer graphics, complex curved surfaces are approximated using thousands of triangular polygons. The interior angle formula (n-2)×180°/n determines how polygon tiles fit together: regular hexagons tessellate perfectly because each 120° interior angle allows exactly 3 hexagons to meet at each vertex (3×120° = 360°). Understanding these relationships prepares students for trigonometry, coordinate geometry, and calculus, where polygon approximations help calculate areas under curves.

§ 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 hexagon have?

Answer: 6

  1. Recall the definition of a hexagon → 6 — A hexagon has 6 sides.
Easy§ 02

What is the name of a 8-sided polygon?

Answer: octagon

  1. Match the number of sides to the polygon name → octagon — A polygon with 8 sides is called a octagon.
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 angles leads to calculating 360°÷6 = 60° for a hexagon's interior angle instead of the correct (6-2)×180°÷6 = 120°.
  • Mixing up the formulas results in using (n-2)×180° for a single interior angle rather than the sum of all interior angles, giving 720° for one hexagon angle instead of 120°.
  • Forgetting that exterior angles sum to 360° for any polygon causes errors like claiming a pentagon's exterior angles sum to 540° instead of 360°.
§ 05

Frequently asked questions

What is the difference between regular and irregular polygons?
Regular polygons have all sides equal in length and all angles equal in measure, like a square or regular hexagon. Irregular polygons have sides or angles that differ in size, such as a rectangle (equal angles but unequal sides) or a scalene triangle (unequal sides and angles).
How do you find the number of sides from an interior angle?
Use the formula n = 360°÷(180° - interior angle). For example, if each interior angle measures 144°, then n = 360°÷(180° - 144°) = 360°÷36° = 10 sides, making it a decagon. This works only for regular polygons.
Why do exterior angles always sum to 360°?
Walking around any polygon's perimeter means turning through one complete rotation, which equals 360°. Each exterior angle represents the amount of turning at each vertex. Regardless of the polygon's shape or number of sides, the total turning remains one full circle.
What happens to polygon angles as the number of sides increases?
Interior angles approach 180° while exterior angles approach 0° as sides increase. A triangle has 60° interior angles, a hexagon has 120°, and a 100-sided polygon has 176.4° interior angles. This is why circles appear to have 180° interior angles everywhere.
How are polygon names determined by their sides?
Polygon names combine Greek or Latin number prefixes with '-gon' meaning angle. Triangle (3), quadrilateral (4), pentagon (5), hexagon (6), heptagon (7), octagon (8), nonagon (9), decagon (10), and dodecagon (12) are the most common. Beyond 12 sides, polygons are often called n-gons.
§ 06

See also

VertexPerimeter
§ 06

Where to next?

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