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types of polygons — Definition, Formula & Examples

Types of polygons are the different ways to classify closed, straight-sided shapes based on their number of sides, whether all sides and angles are equal (regular vs. irregular), and whether all angles point outward (convex vs. concave).

A polygon is a closed plane figure formed by three or more line segments joined end to end. Polygons are classified by the number of sides (triangle, quadrilateral, pentagon, hexagon, etc.), by equality of sides and angles (regular if all sides and all interior angles are congruent, irregular otherwise), and by convexity (convex if every interior angle is less than 180°180°, concave if at least one interior angle exceeds 180°180°).

Key Formula

S=(n−2)×180°S = (n - 2) \times 180°
Where:
  • SS = Sum of interior angles of the polygon
  • nn = Number of sides

How It Works

To identify a polygon's type, first count its sides. A 3-sided polygon is a triangle, 4-sided is a quadrilateral, 5-sided is a pentagon, 6-sided is a hexagon, 7-sided is a heptagon, 8-sided is an octagon, and so on. Next, check whether all sides have the same length and all angles are equal — if so, it is regular. Finally, look at the angles: if every interior angle is less than 180°180°, the polygon is convex; if any angle bends inward past 180°180°, it is concave.

Worked Example

Problem: A shape has 6 equal sides and 6 equal angles. What type of polygon is it, and what is the sum of its interior angles?
Classify by sides: Count the sides: 6 sides makes it a hexagon.
Classify by regularity: All sides are equal and all angles are equal, so it is a regular hexagon.
Find the angle sum: Use the interior angle sum formula with n = 6.
S=(6−2)×180°=4×180°=720°S = (6 - 2) \times 180° = 4 \times 180° = 720°
Answer: It is a regular hexagon with an interior angle sum of 720°720°.

Visualization

Why It Matters

Recognizing polygon types helps you choose the right formulas for area, perimeter, and angle calculations. Architects, game designers, and engineers all select specific polygon types when designing floor plans, 3D models, and structural components.

Common Mistakes

Mistake: Calling any shape with equal sides a regular polygon.
Correction: A polygon is regular only when all sides AND all interior angles are equal. A rhombus has four equal sides but unequal angles, so it is not a regular polygon.

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