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° where n 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 n-gon is a polygon with n 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⋯Vn is a closed piecewise-linear curve ⋃i=1nViVi+1‾ (Vn+1=V1) with non-self-intersecting boundary; interior angle sum =(n−2)π

Worked Examples

Example 1

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

Answer

720°

First step

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

Full solution

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

Example 2

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

Example 3

easy
How many sides does a polygon with interior-angle sum 1080∘ 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∘ for all polygons — the general sum is (n−2)×180∘.

Why This Formula Matters

Polygon is the umbrella that lets one rule — interior angles sum to (n−2)×180∘ — 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 n-gon is a polygon with n 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∘ — 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