# Interior and Exterior Angles

All angles are classified based on whether they are found inside or outside of any geometric shape into interior and exterior angles.

## What are Interior Angles

Angles that are found inside or within any geometric shape are called interior angles. They are also sometimes called internal angles.

A triangle has three interior angles. Similarly a quadrilateral such as a square, rectangle, parallelogram, kite, or a trapezoid has four interior angles. Again polygons such as pentagon, hexagon, heptagon, octagon, nonagon, and decagon have five, six, seven, eight, nine, and ten interior angles.

Shown below are the interior angles of some common regular polygons.

## Formulas

### Sum of Interior Angles

To obtain the sum of interior angles we simply add the measures of all the angles found within the shape. For example, the three angles of a triangle add up to 180°. Similarly quadrilaterals add up to 360°.

To make the process less tedious, the sum of interior angles in all regular polygons is calculated using the formula given below:

Sum of interior angles = (n-2) x 180°, here n = here n = total number of sides

Let us take an example to understand the concept,

For an equilateral triangle, n = 3

Thus,

Sum of interior angles of an equilateral triangle = (n-2) x 180°

= (3-2) x 180°

= 180°

Find the sum of the interior angles of a square.

Solution:

As we know,
Sum of interior angles = (n-2) x 180°, here n = 4
= (4-2) x 180°
= 360°

### One Interior Angle

To find the measure of a single interior angle of a regular polygon, we simply divide the sum of the interior angles value with the total number of sides. For an irregular polygon, the unknown angle can be determined when measure of all other angles and their sum are known.

The formula for determining one interior angle in a regular polygon is given below:

One interior angle = (n-2) x 180°/n, here n = total number of sides

Let us take an example to understand the concept better,

For an equilateral triangle, n = 3

Thus,

One interior angle = (n-2) x 180°/n, here n = 3

= (3-2) x 180°/3

= 60°

Let us take some more examples to understand the concepts better.

Find the measure of one interior angle of a regular dodecagon.

Solution:

As we know,
One interior angle = (n-2) x 180°/n, here n = 12
= (12 – 2) x 180°/12
= 150°

Find the measure of the unknown interior angle in an irregular hexagon with angles 130°, 90°, 140°, 150°, and 90°.

Solution:

As we know,
Sum of interior angles = (n-2) x 180°, here n = 6
= (6-2) x 180°
= 720°
Let the unknown angle be x°
Now,
130° + 90° + 140° + 150° + 90° + x = 720°
x = 720° – 600°
x = 120°

## What are Exterior Angles

Angles that are found outside or external to any geometric shape are called exterior angles. They are also sometimes known as external angles or turning angle. An exterior angle is made by extending one of the lines of the shape beyond the point of intersection.

Shown below are the interior angles of some common regular polygons.

(show only the exterior angles and not interior angles)

Show all the shapes given in the link and mark all their exterior angles. Write the number of interior angles with number alongside the figure.

## Formulas

### Sum of Exterior Angles

Since an exterior angle is formed by extending a side, the sum of the interior and the exterior angle on the same vertex of any polygon is 180°. The formula to determine the sum of exterior angles is derived below:

Now, for any polygon with n sides,

Sum of exterior angles + Sum of interior angles = n x 180°

Thus,

Sum of exterior angles = n x 180° – Sum of all interior angles… (1)

Putting the formula for sum of all interior angles in (1) we get,

Sum of exterior angles = n x 180° – (n-2) x 180°

= n x 180° – (n x 180° + 2 x 180°)

= 180°n – 180°n + 360°

= 360°

Hence,

Sum of the exterior angles of any polygon is 360°.

### One Exterior Angle

To find the measure of a single exterior angle, we simply divide the measure of sum of the exterior angles with the total number of sides. The formula to determine one exterior angle is given below:

One exterior angle = 360°/n, here n = total number of sides

Also the value of an exterior angle can be obtained by subtracting the interior angle from 180°.

One exterior angle = 180° – Adjacent interior angle

Let us take some examples to understand the concept better,

Find the measure of the exterior angle of a decagon when its corresponding interior angle is 144°.

Solution:

As we know,
Exterior angle = 180° – Adjacent interior angle, here interior angle = 144°
= 180° – 144°
=36°

Find the measure of each exterior angle of a regular dodecagon.

Solution:

As we know,
One exterior angle = 360°/n, here n = 12
= 360°/12
= 30°

Each exterior angle of a regular polygon is 18°. Find the number of sides of the polygon

Solution:

As we know,
One exterior angle = 360°/n, here n = total number of sides
Thus,
Total number of sides = 360°/ one exterior angle, here one exterior angle = 18
= 360°/18 = 20°