Measures of Each Interior Angle and Exterior Angle of a Regular Polygon

These QuickNotes give you the most important points to remember for the ESSLCE. They are based on the MoE Grade 9 Mathematics Textbook, Unit 6, Chapter 6.3.

~5 min read

Summary

A regular polygon has all sides equal and all interior angles equal.
Each interior angle of an n-sided regular polygon measures (n − 2) × 180° ÷ n.
Each exterior angle of an n-sided regular polygon measures 360° ÷ n.
To find n from a given angle, work through the exterior angle: n = 360° ÷ exterior angle.

Key Words

Regular polygon: a convex polygon that is both equilateral and equiangular.
Equilateral: having all sides of equal length.
Equiangular: having all interior angles of equal measure.
Each interior angle: the measure of one interior angle of a regular polygon. All of them are the same.
Each exterior angle: the measure of one exterior angle of a regular polygon. All of them are also the same.

What Makes a Polygon Regular?

A polygon is regular when it is equiangular, which means all angles are equal, and equilateral, which means all sides are equal. It must satisfy both conditions at the same time.
The equilateral triangle and the square are the two simplest regular polygons.
A rectangle is equiangular because all four angles are 90°, but its sides are not all equal, so it is not regular.
A rhombus is equilateral because all four sides are equal, but its angles are not all equal, so it is also not regular.
Nature gives us a famous example: bees build their honeycomb from regular hexagons. You can see this in the hives of honey farmers all over Ethiopia.
The Greek mathematician Euclid, who taught around 300 BCE, showed in his book the Elements how to construct an equilateral triangle, a square and a regular pentagon with only a ruler and a compass.

The Two Angle Formulas

From Chapter 6.1, the interior angles of an n-sided polygon add up to (n − 2) × 180°. In a regular polygon all n interior angles are equal, so each one is the sum divided by n.
Each interior angle = (n − 2) × 180° ÷ n.
From Chapter 6.2, the exterior angles add up to 360°. In a regular polygon all n exterior angles are equal, so each one is 360° divided by n.
Each exterior angle = 360° ÷ n.
The two answers always add up to 180°, because the interior and exterior angles at a vertex make a straight line. This gives you a fast shortcut: each interior angle = 180° − 360° ÷ n.

Worked example: find each interior angle and each exterior angle of a regular pentagon.

A pentagon has n = 5 sides.

Each interior angle = (5 − 2) × 180° ÷ 5 = 540° ÷ 5 = 108°.

Each exterior angle = 360° ÷ 5 = 72°.

Check: 108° + 72° = 180°, a straight line, as expected.

Angle Values You Should Know

Regular polygon	Sides (n)	Each interior angle	Each exterior angle
Equilateral triangle	3	60°	120°
Square	4	90°	90°
Pentagon	5	108°	72°
Hexagon	6	120°	60°
Octagon	8	135°	45°
Decagon	10	144°	36°

Working Backwards: Finding n from an Angle

Many exam questions give you an angle and ask for the number of sides. The fastest path always goes through the exterior angle.
Since each exterior angle = 360° ÷ n, we can flip the formula: n = 360° ÷ each exterior angle.
If the question gives an interior angle instead, first subtract it from 180° to get the exterior angle, then divide into 360°.
A regular polygon with a given exterior angle exists only when 360 divided by that angle gives a whole number. For example, no regular polygon has an exterior angle of 50°, because 360 ÷ 50 = 7.2 is not a whole number.

Worked example: each exterior angle of a regular polygon is 72°. How many sides does it have?

n = 360° ÷ 72° = 5.

The polygon is a regular pentagon.

Common Mistakes to Avoid

Using 360° ÷ n for the interior angle. That formula gives the exterior angle. The interior angle is (n − 2) × 180° ÷ n.
Forgetting to divide by n. The expression (n − 2) × 180° alone is the sum of all interior angles, not the measure of each one.
Applying these formulas to a polygon that is not regular. If the sides or angles are not all equal, each angle can be different, and only the sum formulas work.
Calling a rectangle or a rhombus regular. Each of them passes only one of the two tests. A regular polygon needs equal sides and equal angles together.

Easy Ways to Remember

Exterior is easy: 360° ÷ n. Start there, then get the interior angle as 180° minus the exterior angle.
Regular = equal sides + equal angles. Both conditions, always.
Memorize the pentagon pair 108° and 72° and the hexagon pair 120° and 60°. They appear on exams again and again.
To find n from any angle question, reach the exterior angle first, because n = 360° ÷ exterior angle.

Quiz

Tap an answer to check it.

1. What is the measure of each interior angle of a regular hexagon?

2. What is the measure of each exterior angle of a regular octagon?

3. Each interior angle of a regular polygon measures 135°. What is the name of this polygon?

4. In a regular polygon, each interior angle is 5 times each exterior angle. How many sides does the polygon have?

5. If the measure of each exterior angle of a regular polygon is 40°, how many sides does this polygon have?

Remember: A regular polygon has equal sides and equal angles. Each exterior angle is 360° ÷ n, each interior angle is (n − 2) × 180° ÷ n, and the two always add up to 180°. To find n from an angle, go through the exterior angle: n = 360° ÷ exterior angle.