Polygon Theorems

If the number of sides of a polygon is known, we can easily find the sum of its interior angles, the sum of its exterior angles, and the measure of each angle of a regular polygon. Let us prove and apply these formulas step by step.

⭐ Sum of interior angles of an n-sided convex polygon is 2(n − 2) right angles or (n − 2) × 180°

An n-sided convex polygon

Drawing diagonals from one vertex divides the polygon into (n − 2) triangles.

Given:	Let ABCDEFGH ... be an n-sided convex polygon.
To prove:	The sum of its interior angles is 2(n − 2) right angles or (n − 2) × 180°.
Construction:	From vertex A, draw diagonals to all non-adjacent vertices.
Proof:	An n-sided convex polygon is divided into (n − 2) triangles.
∴	Sum of the interior angles of the polygon
=	sum of the interior angles of (n − 2) triangles
=	(n − 2) × 2 right angles [Because the angle sum of a triangle is 2 right angles or 180°]
=	2(n − 2) right angles
=	(n − 2) × 180° [Proved]

⭐ Sum of the exterior angles of a polygon, taken one at each vertex in the same direction, is 360°

Exterior angles of an n-sided polygon

Take one exterior angle at each vertex in the same direction.

Given:	Let ABCDEFG ... be an n-sided convex polygon. At each vertex, take one exterior angle in the same direction.
To prove:	The sum of these n exterior angles is 360°.
Because	At each vertex, an interior angle and its adjacent exterior angle form a linear pair, so their sum is 180°.
∴	The total of all n interior angles and the corresponding n exterior angles is n × 180°.
	The sum of the interior angles is (n − 2) × 180°.
∴	Sum of the exterior angles
=	n × 180° − (n − 2) × 180°
=	2 × 180°
=	360° [Proved]

⭐ What is a regular polygon?

A polygon in which all sides are equal and all interior angles are equal is called a regular polygon.

Regular Polygon

All sides and all interior angles are equal.

👉 A regular triangle is an equilateral triangle, and a regular quadrilateral is a square.

⭐ Each interior angle of a regular n-sided polygon

Because	All interior angles of a regular polygon are equal.
	The sum of the interior angles of an n-sided polygon is (n − 2) × 180°.
∴	Each interior angle
=

⭐ Each exterior angle of a regular n-sided polygon

Because	All exterior angles of a regular polygon are equal, and their sum is 360°.
∴	Each exterior angle

⭐ Solved Example 1: Find the sum of the interior angles of a 12-sided polygon

For an n-sided polygon,

Sum = (n − 2) × 180°

Here, n = 12.

= (12 − 2) × 180° = 10 × 180° = 1800°

Therefore, the sum of the interior angles is 1800°.

⭐ Solved Example 2: Find each interior and exterior angle of a regular octagon

For a regular octagon, n = 8.

Each exterior angle:

Each interior angle:

180° − 45° = 135°

Therefore, each interior angle is 135° and each exterior angle is 45°.

⭐ Solved Example 3: A regular polygon has each exterior angle equal to 30°. Find the number of sides.

For a regular polygon,

Therefore, the polygon has 12 sides.

📝 Practice Problems

Find the sum of the interior angles of a hexagon.
Find the sum of the interior angles of a decagon.
Each exterior angle of a regular polygon is 40°. Find the number of sides.
Find each interior angle of a regular nonagon.
The sum of the interior angles of a polygon is 1260°. Find its number of sides.
Can a regular polygon have each exterior angle equal to 28°? Give a reason.

Show answers

720°
1440°
9 sides
140°
9 sides
No. 360 ÷ 28 is not a whole number, so such a regular polygon is not possible.

📚 Important points to remember

👉 Sum of interior angles of an n-sided polygon = (n − 2) × 180°.

👉 Sum of one exterior angle at each vertex, taken in the same direction, is always 360°.

👉 Each interior angle of a regular polygon is .

👉 Each exterior angle of a regular polygon is .

👉 An interior angle and its adjacent exterior angle add up to 180°.

👉 If each exterior angle of a regular polygon is known, then .