Exterior Angle of a Polygon: Formula, Rules and Examples

Exterior Angle of a Polygon is the angle formed when one side is extended at a vertex. For a convex polygon, the sum of one exterior angle at each vertex is 360°. In a regular polygon, each exterior angle equals 360° divided by the number of sides. These rules help solve direct and reverse angle problems.

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What Is the Exterior Angle of a Polygon?

An exterior angle of a polygon is the angle between one side and the extension of an adjacent side. For a convex polygon, the sum of one exterior angle at every vertex, measured in the same direction, is 360°.

At any vertex, the interior angle and its adjacent exterior angle form a linear pair, so their sum is 180°. Therefore, exterior angle = 180° − interior angle. For a regular polygon with n sides, all exterior angles are equal and each one measures 360°/n.

Exterior Angle 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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Exterior Angle of a Polygon Formula & Tricks

Important Formulas

Sum of exterior angles
Sum = 360°

For any convex polygon, one exterior angle is taken at each vertex in the same direction.

Exterior angle of a regular polygon
Each exterior angle = 360°/n

Here, n is the number of sides of the regular polygon.

Relation with the interior angle
Exterior angle = 180° − interior angle

An interior angle and its adjacent exterior angle form a straight angle.

Number of sides from an exterior angle
n = 360°/e

For a regular polygon, e is the measure of each exterior angle.

Exterior angle from the number of sides
e = 360°/n

This gives the equal exterior angle of a regular n-sided polygon.

Quick Tricks

Use 360° directly for a regular polygon

If the polygon is regular, divide 360° by its number of sides instead of finding the interior angle first.

Example: A regular octagon has exterior angle 360°/8 = 45°.
Find the number of sides by division

When each exterior angle of a regular polygon is given, divide 360° by that angle.

Example: If each exterior angle is 24°, then n = 360°/24° = 15 sides.
Convert from the interior angle

For adjacent interior and exterior angles, subtract the interior angle from 180°.

Example: If the interior angle is 135°, the exterior angle is 180° − 135° = 45°.

Exterior Angle of a Polygon Concepts

Sum of Exterior Angles of a Polygon

The sum of one exterior angle at each vertex of a convex polygon is always 360°.

This result does not depend on the number of sides. The exterior angles represent one complete turn around the polygon. If some exterior angles are known, an unknown angle can be found by subtracting their sum from 360°.

Example: If three exterior angles are 80°, 95° and 110° in a polygon, the fourth angle is 360° − (80° + 95° + 110°) = 75°.

Exterior Angle of a Regular Polygon

Each exterior angle of a regular n-sided polygon is 360°/n.

A regular polygon has equal sides and equal angles, so all its exterior angles are equal. Increasing the number of sides makes each exterior angle smaller.

Example: For a regular decagon, each exterior angle = 360°/10 = 36°.

Relation Between Interior and Exterior Angles

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

The exterior angle is formed by extending one side of the polygon. Since the two angles form a linear pair, e = 180° − I and I = 180° − e, where I is the interior angle and e is the exterior angle.

Example: For an interior angle of 150°, the adjacent exterior angle is 180° − 150° = 30°.

Finding the Number of Sides

For a regular polygon, the number of sides is n = 360°/e.

The result must be a positive whole number because a polygon has a whole number of sides. If the given angle does not divide 360° exactly, it may represent a non-regular polygon or an unsuitable value for this formula.

Example: If each exterior angle is 30°, then n = 360°/30° = 12, so the polygon is a regular dodecagon.

Exterior Angle of a Polygon Visual Guide & Data

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

Exterior Angles of Common Regular Polygons

Each exterior angle is calculated using 360°/n.

Polygon Number of sides (n) Each exterior angle
Equilateral triangle 3 120°
Square 4 90°
Regular pentagon 5 72°
Regular hexagon 6 60°
Regular octagon 8 45°

Note: The exterior angle values in the last column add to 360° when multiplied by the corresponding number of sides.

Exterior Angle of a Regular Pentagon

Visual guide: Exterior Angle of a Regular Pentagon
Number of sides: n = 5 Each exterior angle: 360°/5 = 72° Each interior angle: 180° − 72° = 108°
A regular pentagon has five equal exterior angles, each measuring 72°, and their total is 360°.

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Practice Exterior Angle of a Polygon Questions

Practise published questions related to this topic.

1Which of the following numbers is divisible by 8?→ 2Simplify the given expression. 12 + {6 × 27 ÷ 3 + 2} – (16 ÷ 8) × 5→ 3If A = 4X and B = 9X, then B is what percent of A?→ 4A shopkeeper bought 10 kg wheat at ₹ 20 per kg and 15 kg wheat at ₹ 30 per kg and mixed them. At what price per kg should he sell the mixed variety of wheat to earn 10% profit ?→ 5A dealer purchased an AC for ₹1,848. After allowing a discount of 58% on its marked price, he still gains 63%. Find the marked price of the AC.→ 6A shopkeeper bought certain goods for ₹2,500 and sold them for ₹3,250. Find his profit percentage.→ 7(√32 + √162)/√18 = ?→ 8If the HCF of three numbers ‘x’, 153 and 102 is 17, then the number ‘x’ CANNOT be:→ 9What is the remainder when 1200 is divided by 9?→ 10A trader lists a shirt for ₹ 2,000 and gives a 10% discount to a buyer. Despite the discount, he still gains a profit of 20%. The shirt's cost price→

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Quick Revision Notes

Exterior Angle of a Polygon: Quick Revision

Use these formulas and rules to solve exterior-angle calculations.

  • The sum of one exterior angle at each vertex of a convex polygon is 360°.
  • For a regular n-gon, each exterior angle is 360°/n.
  • An adjacent interior angle and exterior angle sum to 180°.
  • Exterior angle = 180° − interior angle.
  • Number of sides of a regular polygon = 360°/each exterior angle.
  • A regular polygon has equal exterior angles.

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Exterior Angle of a Polygon FAQs

What is the exterior angle of a regular polygon with 12 sides?

Each exterior angle = 360°/12 = 30°.

What is the sum of the exterior angles of a hexagon?

The sum is always 360° for a convex polygon, so the exterior angles of a hexagon add up to 360°.

A regular polygon has an exterior angle of 20°. How many sides does it have?

n = 360°/20° = 18. Therefore, it is a regular 18-sided polygon.

If the interior angle of a polygon is 125°, what is the adjacent exterior angle?

Exterior angle = 180° − 125° = 55°.

What is the exterior angle of a regular pentagon?

A regular pentagon has 5 sides, so each exterior angle = 360°/5 = 72°.

Can the formula 360°/n be used for every polygon?

It gives each exterior angle only when the polygon is regular. For an irregular polygon, the exterior angles may differ, but their total is still 360° for a convex polygon.

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