Polygon Formula

Polygon is a closed two-dimensional figure formed by three or more straight line segments connected end-to-end.

The Formula

Interior angle sum=(n−2)×180°\text{Interior angle sum} = (n-2) \times 180° where nn is the number of sides

When to use: Connect-the-dots that closes into a shape—no curves allowed.

Quick Example

Triangle (3 sides), quadrilateral (4), pentagon (5), hexagon (6)...

Notation

An nn-gon is a polygon with nn sides; regular means all sides and angles are equal

What This Formula Means

A closed two-dimensional figure formed by three or more straight line segments connected end-to-end.

Connect-the-dots that closes into a shape—no curves allowed.

Formal View

A polygon Pn=V1V2⋯VnP_n = V_1 V_2 \cdots V_n is a closed piecewise-linear curve ⋃i=1nViVi+1‾\bigcup_{i=1}^n \overline{V_i V_{i+1}} (Vn+1=V1V_{n+1} = V_1) with non-self-intersecting boundary; interior angle sum =(n−2)π= (n-2)\pi

Worked Examples

Example 1

easy
What is the sum of the interior angles of a hexagon?

Answer

720°720°

First step

1
Step 1: Use the interior angle sum formula: (n−2)×180°(n-2) \times 180° where nn is the number of sides.

Full solution

  1. 2
    Step 2: A hexagon has n=6n = 6 sides.
  2. 3
    Step 3: Sum =(6−2)×180°=4×180°=720°= (6-2) \times 180° = 4 \times 180° = 720°.
Any polygon can be divided into (n−2)(n-2) triangles by drawing diagonals from one vertex. Since each triangle contributes 180°180°, the total interior angle sum is (n−2)×180°(n-2) \times 180°. For a hexagon, this gives 720°720°.

Example 2

medium
Each interior angle of a regular polygon is 150°150°. How many sides does it have?

Example 3

easy
How many sides does a polygon with interior-angle sum 1080∘1080^\circ have?

Common Mistakes

  • Calling a figure with a curved edge a polygon — every side must be a straight segment.
  • Counting an open chain as a polygon — a polygon must close back to its starting point.
  • Using the triangle's 180∘180^\circ for all polygons — the general sum is (n−2)×180∘(n-2)\times 180^\circ.

Why This Formula Matters

Polygon is the umbrella that lets one rule — interior angles sum to (n−2)×180∘(n-2)\times 180^\circ — cover every straight-sided shape at once, turning many separate facts (triangle 180°, quadrilateral 360°) into a single formula. Recognizing it by "Is the figure closed and made of three or more straight sides with no curves?" — rather than by familiar numbers — is what lets a student tell it apart from circle and triangle and open figure in a mixed problem set.

Frequently Asked Questions

What is the Polygon formula?

A closed two-dimensional figure formed by three or more straight line segments connected end-to-end.

How do you use the Polygon formula?

Connect-the-dots that closes into a shape—no curves allowed.

What do the symbols mean in the Polygon formula?

An nn-gon is a polygon with nn sides; regular means all sides and angles are equal

Why is the Polygon formula important in Math?

Polygon is the umbrella that lets one rule — interior angles sum to (n−2)×180∘(n-2)\times 180^\circ — cover every straight-sided shape at once, turning many separate facts (triangle 180°, quadrilateral 360°) into a single formula. Recognizing it by "Is the figure closed and made of three or more straight sides with no curves?" — rather than by familiar numbers — is what lets a student tell it apart from circle and triangle and open figure in a mixed problem set.

What do students get wrong about Polygon?

The procedure for polygon is the easy part; the trap is calling a figure with a curved edge a polygon. Asking "Is the figure closed and made of three or more straight sides with no curves?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Polygon formula?

Before studying the Polygon formula, you should understand: line, angles.

← Back to Polygon See All Examples Practice Problems

Related Concepts

LineAngles