Polygon Formula

The Formula

\text{Interior angle sum} = (n-2) \times 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 P_n = V_1 V_2 \cdots V_n is a closed piecewise-linear curve \bigcup_{i=1}^n \overline{V_i V_{i+1}} (V_{n+1} = V_1) with non-self-intersecting boundary; interior angle sum = (n-2)\pi

Worked Examples

Example 1

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

Solution

  1. 1
    Step 1: Use the interior angle sum formula: (n-2) \times 180° where n is the number of sides.
  2. 2
    Step 2: A hexagon has n = 6 sides.
  3. 3
    Step 3: Sum = (6-2) \times 180° = 4 \times 180° = 720°.

Answer

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) \times 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?

Common Mistakes

  • Including shapes with curved sides as polygons — polygons must have only straight edges
  • Confusing regular polygons (all sides and angles equal) with irregular ones
  • Forgetting that polygons must be closed — an open chain of segments is not a polygon

Why This Formula Matters

Foundation for understanding all 2D closed figures with straight edges.

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?

Foundation for understanding all 2D closed figures with straight edges.

What do students get wrong about Polygon?

Convex polygons have no indentations; concave polygons have at least one vertex that points inward.

What should I learn before the Polygon formula?

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

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Related Concepts

LineAngles