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Interior & Exterior Angles

Every polygon carries two angle totals: the interior angles that grow with the number of sides, and the exterior angles that always come to 360. Two neat formulas handle both.

MYP 4Standard MathsGeometryCriteria A · C · D~10 min read

Written by Fayek Jawad Haider · MYP student

Interior angle sum

An interior angle is an angle inside the polygon at one of its corners. To find the total of all of them, split the polygon into triangles from one corner. A polygon with \(n\) sides splits into \(n - 2\) triangles, and each triangle gives \(180^\circ\).

\[ \text{interior angle sum} = (n - 2) \times 180^\circ \]
Worked example

Find the sum of the interior angles of a hexagon (6 sides).

1
Use the formula with \(n = 6\): sum \(= (6 - 2) \times 180\).
2
Work the bracket first: \(6 - 2 = 4\).
3
Multiply: \(4 \times 180 = 720\).
The interior angles add up to \(720^\circ\)

It is \(n - 2\), not \(n\)

The most common slip is multiplying \(n \times 180\). Always subtract 2 from the number of sides first, because a polygon splits into two fewer triangles than it has sides.

Exterior angles sum to 360

An exterior angle is the angle between one side and the extension of the next side. Whatever the polygon, walking once around the outside turns you through one full circle, so the exterior angles always add to \(360^\circ\).

\[ \text{exterior angle sum} = 360^\circ \]

At any single corner the interior and exterior angles sit on a straight line together, so:

\[ \text{interior angle} + \text{exterior angle} = 180^\circ \]

Regular polygons

In a regular polygon all sides and all angles are equal. Because the exterior angles are equal and total \(360^\circ\), each one is simply \(360\) shared between the sides.

\[ \text{each exterior angle} = \frac{360^\circ}{n} \]
Worked example

Find each interior angle of a regular hexagon (6 sides).

1
Each exterior angle is \(360 \div 6 = 60^\circ\).
2
Interior and exterior angles add to \(180^\circ\), so each interior angle is \(180 - 60\).
Each interior angle is \(120^\circ\)

You can check this against the sum: \(720 \div 6 = 120^\circ\), which agrees.

Finding the number of sides

Turn the exterior angle formula around. If you know one exterior angle of a regular polygon, then \(n = 360 \div \text{exterior angle}\). If you are given the interior angle instead, find the exterior angle first by subtracting from \(180^\circ\).

Worked example

A regular polygon has an interior angle of \(150^\circ\). How many sides does it have?

1
Find the exterior angle: \(180 - 150 = 30^\circ\).
2
Divide \(360\) by the exterior angle: \(n = 360 \div 30\).
The polygon has 12 sides

Where this is assessed

Recalling and using the two formulas is Criterion A (Knowing and understanding). Laying out the working so each figure is justified is Criterion C (Communicating). Working backwards from an angle to the number of sides, or explaining which regular shapes tile a floor, is Criterion D (Applying maths in context).

Check yourself

1. Find the sum of the interior angles of a pentagon (5 sides). +

Use \((n - 2) \times 180\) with \(n = 5\): \((5 - 2) \times 180 = 3 \times 180 = \mathbf{540^\circ}\).

2. Find each exterior angle and each interior angle of a regular octagon (8 sides). +

Each exterior angle is \(360 \div 8 = \mathbf{45^\circ}\). Each interior angle is \(180 - 45 = \mathbf{135^\circ}\).

3. A regular polygon has an interior angle of \(140^\circ\). How many sides does it have? +

The exterior angle is \(180 - 140 = 40^\circ\), so \(n = 360 \div 40 = \mathbf{9}\) sides.

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