Polygon Angle Sum Formula

Learn the formula to calculate the sum of interior angles in any polygon.

Intermediate25 minLesson
Printable worksheet

Definition

The interior angle sum of a polygon is the total of all its interior angles added together.

The Formula

For any polygon with nn sides:
Sum of interior angles=(n−2)×180°\text{Sum of interior angles} = (n - 2) \times 180°

Why (n−2)(n - 2)?

Any polygon can be divided into triangles by drawing diagonals from one vertex. A polygon with nn sides can be divided into exactly (n−2)(n - 2) triangles. Since each triangle has angles summing to 180°180°, we multiply.

Examples

PolygonSides (nn)TrianglesAngle Sum
Triangle31180°180°
Quadrilateral42360°360°
Pentagon53540°540°
Hexagon64720°720°

Try it now

What is the sum of the interior angles of a triangle?

Worked Examples

What is the sum of the interior angles of a pentagon?
1

Identify the number of sides

A pentagon has 5 sides, so n=5n = 5
2

Apply the formula

Sum =(n−2)×180°= (n - 2) \times 180° → Sum =(5−2)×180°= (5 - 2) \times 180°
3

Calculate the number of triangles

5−2=35 - 2 = 3 triangles
4

Multiply by 180°

3×180°=540°3 \times 180° = 540°

Common Mistakes

Using n×180°n \times 180° instead of (n−2)×180°(n - 2) \times 180°

Why it's wrong: Students forget to subtract 2 from the number of sides. The formula counts triangles, not sides.

Correct: Always subtract 2 first: (n−2)×180°(n - 2) \times 180°. A pentagon has 5 sides but only 3 triangles.

Confusing interior angles with exterior angles

Why it's wrong: The exterior angle sum is always 360°360° for any convex polygon, regardless of the number of sides.

Correct: Interior angles are inside the polygon. The formula (n−2)×180°(n - 2) \times 180° is for interior angles only.

Forgetting that triangles sum to 180°180°

Why it's wrong: The whole formula is based on dividing the polygon into triangles.

Correct: Remember: triangle = 180°180°, that's why we multiply (n−2)(n - 2) by 180°180°.

Interactive Visual

Polygon explorer

3 triangles × 180°

Triangles from one vertex

5−2=35 - 2 = 3

Sum of the interior angles

(5−2)⋅180∘=540∘(5 - 2) \cdot 180^\circ = 540^\circ

Each interior angle (regular polygon)

540∘÷5=108∘540^\circ \div 5 = 108^\circ

The diagonals from one vertex cut the polygon into n − 2 triangles, each with 180°.

Triangle Explorer

By sides:Isosceles
By angles:Acute

Area:

20,000.0 sq units

Explore the relationship between angles in a triangle.

Interactive Sandbox

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Practice Problems

15 problems
Problem 1 of 15
Easy
What is the sum of the interior angles of a triangle?

Why It Matters

The polygon angle sum formula is essential for:
  • Architecture: Designing buildings with polygonal rooms or windows
  • Engineering: Calculating stress points in truss structures
  • Art and Design: Creating tessellations and geometric patterns
  • Navigation: Understanding direction changes in multi-leg routes
  • Game Development: Programming collision detection for polygonal shapes
Once you know this formula, you can find missing angles in any polygon!

Real World Applications

Architecture and Design

Architects use the angle sum formula when designing polygonal rooms, windows, or floor plans.

Example:

A pentagonal window has 5 sides. The angle sum is (5−2)×180°=540°(5 - 2) \times 180° = 540°. If 4 corners are right angles (90° each), the fifth angle must be 540°−360°=180°540° - 360° = 180°... which is impossible! This tells the architect the design needs adjustment.

1Try It Yourself

You're designing an octagonal gazebo. Each corner needs an equal angle for symmetry.

What angle should each corner be?

Step 1: Write the mathematical expression

First find the total: (8−2)×180(8 - 2) \times 180, then divide by 8

Sports Field Design

Sports facilities sometimes use polygonal shapes for efficiency or aesthetics.

Example:

A hexagonal practice area has interior angles summing to 720°720°. If it's regular, each angle is 720°÷6=120°720° \div 6 = 120°.

2Try It Yourself

A heptagonal (7-sided) training zone is being painted. You need to know the angle sum to place equipment correctly.

What is the sum of interior angles?

Step 1: Write the mathematical expression

Use the formula: (n−2)×180(n - 2) \times 180

Key Takeaways

  • 1The interior angle sum of a polygon with nn sides is (n−2)×180°(n - 2) \times 180°
  • 2The formula works because any polygon can be divided into (n−2)(n - 2) triangles
  • 3Triangle = 180°180°, Quadrilateral = 360°360°, Pentagon = 540°540°, Hexagon = 720°720°
  • 4In a regular polygon, each angle equals the total sum divided by the number of angles
  • 5To find a missing angle, subtract the known angles from the total sum

Frequently Asked Questions

Yes, for all simple (non-self-intersecting) polygons, whether regular or irregular, convex or concave.
Yes, for all simple (non-self-intersecting) polygons, whether regular or irregular, convex or concave.
When you draw diagonals from one vertex, you create (n−2)(n - 2) triangles. Try it with a quadrilateral: 2 triangles. Pentagon: 3 triangles. The pattern is always n−2n - 2.
The sum of exterior angles is always 360°360° for any convex polygon, regardless of the number of sides. This is a separate formula!

Glossary

Interior angle
An angle formed inside a polygon between two adjacent sides
Polygon
A closed 2D shape with straight sides (e.g., triangle, quadrilateral, pentagon)
Regular polygon
A polygon where all sides and all angles are equal
Diagonal
A line segment connecting two non-adjacent vertices of a polygon
Convex polygon
A polygon where all interior angles are less than 180°180°
Concave polygon
A polygon that has at least one interior angle greater than 180°180°

Formula Card

Interior Angle Sum

(n−2)×180°(n - 2) \times 180°

Sum of all interior angles in a polygon with n sides

Each Angle (Regular)

(n−2)×180°n\frac{(n - 2) \times 180°}{n}

Each interior angle in a regular polygon with n sides

Today's challengeOne problem, three tries

More in This Topic

Introduction to Polygons

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Diagonals of Polygons

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Classifying Polygons

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Exterior Angles of Polygons

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