Sum of Interior Angles of a Polygon with Formula Examples

Sum of Interior Angles of a Polygon is found using the number of its sides. An n-sided polygon can be divided into n − 2 triangles from one vertex, so the angle sum is (n − 2) × 180°. This page covers the formula, regular polygon angles, reverse calculations and solved aptitude problems.

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What Is the Sum of Interior Angles of a Polygon?

The sum of the interior angles of an n-sided polygon is (n − 2) × 180°.

An interior angle lies inside the boundary of a polygon. Drawing diagonals from one vertex divides a polygon into n − 2 triangles. Since each triangle has an angle sum of 180°, the polygon's total interior angle sum is (n − 2) × 180°. For a hexagon, the sum is (6 − 2) × 180° = 720°.

Sum of Interior Angles of a Polygon Video Lessons

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Polygons: Angle Sum and Exterior Angle

Learn the formulas for the sum of interior angles and exterior angles of polygons, with guidance on applying them to solve geometry problems.

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Sum of Interior Angles of a Polygon Formula & Tricks

Important Formulas

Interior angle sum
S = (n − 2) × 180°

Here, S is the sum of all interior angles and n is the number of sides.

Number of sides from angle sum
n = S ÷ 180° + 2

Use this rearranged formula when the total interior angle sum is given.

Each interior angle of a regular polygon
Each angle = [(n − 2) × 180°] ÷ n

All interior angles of a regular polygon are equal.

Each exterior angle of a regular polygon
Each exterior angle = 360° ÷ n

The exterior angles of a regular polygon have a total of 360°.

Quick Tricks

Use triangles to remember the formula

A polygon with n sides forms n − 2 triangles when diagonals are drawn from one vertex. Multiply the number of triangles by 180°.

Example: A pentagon forms 5 − 2 = 3 triangles, so its interior angle sum is 3 × 180° = 540°.
Check a regular polygon using exterior angles

For a regular polygon, calculate one exterior angle as 360° ÷ n. Subtract it from 180° to get one interior angle.

Example: For a regular octagon, one exterior angle is 360° ÷ 8 = 45°, so one interior angle is 180° − 45° = 135°.
Reverse the formula directly

If the total angle sum is known, divide it by 180° and add 2 to obtain the number of sides.

Example: If the sum is 1,260°, then n = 1,260 ÷ 180 + 2 = 7 + 2 = 9 sides.

Sum of Interior Angles of a Polygon Concepts

Triangulation of a Polygon

An n-sided polygon can be divided into n − 2 triangles by drawing diagonals from one vertex.

Each triangle contributes 180° to the interior angle sum. Therefore, the total is (n − 2) × 180°. A quadrilateral forms 2 triangles, giving an angle sum of 2 × 180° = 360°.

Example: A heptagon forms 7 − 2 = 5 triangles. Its interior angle sum is 5 × 180° = 900°.

Interior Angle Sum by Number of Sides

The angle sum increases by 180° whenever one side is added to a polygon.

The formula S = (n − 2) × 180° applies to triangles, quadrilaterals, pentagons and all simple polygons. It gives 180° for a triangle, 360° for a quadrilateral and 540° for a pentagon.

Example: For a decagon, S = (10 − 2) × 180° = 8 × 180° = 1,440°.

Interior Angles of a Regular Polygon

In a regular polygon, every interior angle has the same measure.

First find the total angle sum using (n − 2) × 180°, then divide by n. Equivalently, one interior angle equals 180° − 360° ÷ n.

Example: For a regular hexagon, each angle is [(6 − 2) × 180°] ÷ 6 = 720° ÷ 6 = 120°.

Finding the Number of Sides

If the total interior angle sum is S, the polygon has n = S ÷ 180° + 2 sides.

The given sum must be at least 180° and should produce an integer value of n for a standard polygon. For S = 1,080°, n = 1,080 ÷ 180 + 2 = 8 sides.

Example: A polygon has an interior angle sum of 1,620°. Its number of sides is 1,620 ÷ 180 + 2 = 9 + 2 = 11.

Sum of Interior Angles of a Polygon Visual Guide & Data

Use the table and chart below to understand the data better.

Interior Angle Sums for Common Polygons

The total interior angle sum calculated using S = (n − 2) × 180°.

Polygon Number of sides (n) Interior angle sum
Triangle 3 180°
Quadrilateral 4 360°
Pentagon 5 540°
Hexagon 6 720°
Octagon 8 1,080°
Decagon 10 1,440°

Note: These are total angle sums. For a regular polygon, divide the total by the number of sides to find one interior angle.

Triangulating a Hexagon

Visual guide: Triangulating a Hexagon
Number of sides: n = 6 Triangles formed: n − 2 = 4 Interior angle sum: 4 × 180° = 720°
A hexagon can be divided into four triangles from one vertex, so its interior angle sum is 4 × 180° = 720°.

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Practice Sum of Interior Angles of a Polygon Questions

Practise published questions related to this topic.

1The price of petrol is increased by 28%. A person wants to increase his expenditure by 22% only. By approximately what percent should he decrease his consumption?→ 2An article is sold for Rs. 288 after successive discounts of 20% and 25%. What is the marked price of the article?→ 321.6 ÷ 3.6 × 2 + 0.25 × 16 ÷ 4 - 6 is equal to∶→ 4What is the value of x so that the seven digit number 55350x2 is divisible by 72?→ 5Two articles are sold for Rs. 4,752 each. On one, the seller gains 32% and on the other he loses 28%. What is his overall gain or loss percentage, correct to one decimal place?→ 6The price of sugar is increased by 17%. A person wants to increase his expenditure by 5% only. By approximately what percent should he decrease his consumption?→ 7The value of 15.2 + 5.8 ÷ 2.9 × 2 - 3.5 × 2 ÷ 0.5 is equal to:→ 8What is the value of x so that the seven digit number 8439x53 is divisible by 99?→ 9Two articles are sold for Rs. 962 each. On one, the seller gains 30% and on the other he loses 26%. What is his overall gain or loss percentage, nearest to one decimal place?→ 10An article is sold for Rs. 288 after successive discounts of 25% and x%. If the marked price of the article is Rs. 480, what is the value of x?→

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Quick Revision Notes

Sum of Interior Angles of a Polygon: Revision Points

Use these direct rules for quick calculations.

  • For n sides, total interior angle sum = (n − 2) × 180°.
  • The number of triangles formed from one vertex is n − 2.
  • Number of sides = total angle sum ÷ 180° + 2.
  • For a regular polygon, each interior angle = [(n − 2) × 180°] ÷ n.
  • For a regular polygon, each exterior angle = 360° ÷ n.
  • One interior angle and its corresponding exterior angle add up to 180°.

Sum of Interior Angles of a Polygon FAQs

What is the sum of the interior angles of a polygon with 12 sides?

S = (12 − 2) × 180° = 10 × 180° = 1,800°.

How many sides does a polygon have if its interior angle sum is 900°?

n = 900° ÷ 180° + 2 = 5 + 2 = 7. The polygon is a heptagon.

What is the sum of the interior angles of a regular pentagon?

The total sum is (5 − 2) × 180° = 540°. Each angle of a regular pentagon is 540° ÷ 5 = 108°.

What is the interior angle sum of a quadrilateral?

A quadrilateral has 4 sides, so its interior angle sum is (4 − 2) × 180° = 360°.

What is the difference between total interior angle sum and one interior angle?

The total interior angle sum includes all n interior angles and equals (n − 2) × 180°. One interior angle can be found by dividing this total by n only when the polygon is regular.

Can the interior angle sum of a polygon be less than 180°?

No. The smallest polygon is a triangle, whose interior angle sum is 180°. A polygon with more sides has a greater sum.

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