Interactive Polygon Calculator

Enter the number of sides ($n$) below to see the full breakdown of properties for that polygon.

Polygon Calculator

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Key Formulas for Polygons

For any polygon with $n$ sides, here are the essential formulas you need to know for Secondary 1 & 2 Math.

1. Sum of Interior Angles

The sum of all interior angles in any polygon (regular or irregular) is found by dividing the polygon into triangles.

$$\text{Sum of Interior Angles} = (n - 2) \times 180^\circ$$

Why? A polygon with $n$ sides can be divided into $(n-2)$ triangles from one vertex. Since each triangle has angles adding up to $180^\circ$, the total sum is $(n-2) \times 180^\circ$.

2. Each Interior Angle (Regular Polygon)

For a regular polygon (where all sides and angles are equal), you can find the size of one interior angle by dividing the sum by $n$.

$$\text{Interior Angle} = \frac{(n - 2) \times 180^\circ}{n}$$

3. Each Exterior Angle (Regular Polygon)

The sum of exterior angles of any convex polygon is always $360^\circ$. For a regular polygon, each exterior angle is the same size.

$$\text{Exterior Angle} = \frac{360^\circ}{n}$$

4. Number of Diagonals

A diagonal is a line segment connecting two non-adjacent vertices.

$$\text{Number of Diagonals} = \frac{n(n - 3)}{2}$$

Worked Examples

Example 1: Finding Angle Sum

Question: Find the sum of interior angles of a decagon (10 sides).

Solution: Using the formula with $n = 10$: $\text{Sum} = (10 - 2) \times 180^\circ$ $\text{Sum} = 8 \times 180^\circ$ $\text{Sum} = 1440^\circ$

Example 2: Finding Number of Sides

Question: Each exterior angle of a regular polygon is $24^\circ$. How many sides does it have?

Solution: Using the exterior angle formula: $\text{Exterior Angle} = \frac{360^\circ}{n}$ $24^\circ = \frac{360^\circ}{n}$ $n = \frac{360^\circ}{24^\circ}$ $n = 15$ The polygon has 15 sides.

Polygon Names Reference

Sides ($n$)	Name	Interior Sum	Each Interior (Regular)
3	Triangle	$180^\circ$	$60^\circ$
4	Quadrilateral	$360^\circ$	$90^\circ$
5	Pentagon	$540^\circ$	$108^\circ$
6	Hexagon	$720^\circ$	$120^\circ$
7	Heptagon	$900^\circ$	$\approx 128.57^\circ$
8	Octagon	$1080^\circ$	$135^\circ$
9	Nonagon	$1260^\circ$	$140^\circ$
10	Decagon	$1440^\circ$	$144^\circ$
12	Dodecagon	$1800^\circ$	$150^\circ$