In a polygon, the angles measured inside the polygon are called its interior angles.

A polygon can be classified into two types, namely:

In a regular polygon all the sides are of the same measure, and all the interior angles are equal. However, in an irregular polygon the measure of the side lengths and interior angles may have different values.

We can use the following formula to calculate the sum of interior angles of a polygon:

Sum of interior angles of a polygon, \(S=(n-2)\times 180^{\circ}\).

We will discuss this formula in detail in the next section.

To calculate the sum of interior angles of a polygon, you need its number of sides.

Mathematically, the sum of the angles is expressed as:

Sum of interior angles of a polygon \(=(n-2)\times 180^{\circ}\)

Where, n is the number of sides of the polygon

The angle is measured in degrees.

The sum of interior angles of a polygon formula can be used to measure:

[Note: A polygon with n number of sides can be referred to as a “n-gon”]

Example 1: Find the sum of interior angles in the STOP sign.

Solution:

The sign is in the shape of a hexagon.

It has 6 sides.

Sum of interior angles, \(S=(n-2)\times 180^{\circ}\) [Write the formula]

\(S=(6-2)\times 180^{\circ}\) [Substitute 6 for n]

\(S=(4)\times 180^{\circ}\) [Subtract]

\(S=720^{\circ}\) [Multiply]

Hence, the sum of interior angles is \(720^{\circ}\).

Example 2: Find the sum of interior angles for the irregular polygon.

Solution:

The figure is in the shape of an irregular pentagon.

It has 5 sides.

Sum of interior angles, \(S=(n-2)\times 180^{\circ}\) [Write the formula]

\(S=(5-2)\times 180^{\circ}\) [Substitute 5 for n]

\(S=(3)\times 180^{\circ}\) [Subtract]

\(S=540^{\circ}\) [Multiply]

Hence, the sum of interior angles is \(540^{\circ}\).

Example 3: What is the sum of the interior angles of the medal shown in the figure.

Solution:

The medal has the shape of a decagon

It has 10 sides.

Sum of interior angles, \(S=(n-2)\times 180^{\circ}\) [Write the formula]

\(S=(10-2)\times 180^{\circ}\) [Substitute 10 for n]

\(S=(8)\times 180^{\circ}\) [Subtract]

\(S=1440^{\circ}\) [Multiply]

Hence, the sum of interior angles is \(1440^{\circ}\).

Example 4: Find the measure of each interior angle of a regular octagon.

Solution:

An octagon has 8 sides. Let us first determine the sum of interior angles:

Sum of interior angles, \(S=(n-2)\times 180^{\circ}\) [Write the formula]

\(S=(8-2)\times 180^{\circ}\) [Substitute 8 for n]

\(S=(6)\times 180^{\circ}\) [Subtract]

\(S=1080^{\circ}\) [Multiply]

In a regular polygon, all interior angles are equal in measure. So we can find each interior angle by dividing the sum of interior angles by the number of sides.

\( \frac{\text{Sum of interior angles}}{\text{Number of sides}} \)

\( =\frac{1080^{\circ}}{8} \) [Substitute the values]

\( =135^{\circ} \) [Divide]

Hence, each interior angle measures \( 135^{\circ} \).

Example 5: Find the measure of each interior angle of a regular 15-gon.

Solution:

A 15-gon has 15 sides. Let us first determine the sum of interior angles:

Sum of interior angles, \(S=(n-2)\times 180^{\circ}\) [Write the formula]

\(S=(15-2)\times 180^{\circ}\) [Substitute 15 for n]

\(S=(13)\times 180^{\circ}\) [Subtract]

\(S=2340^{\circ}\) [Multiply]

In a regular polygon all interior angles are equal in measure. So we can find each interior angle by dividing the sum of interior angles by the number of sides.

\( \frac{\text{Sum of interior angles}}{\text{Number of sides}} \)

\( =\frac{2340^{\circ}}{15} \) [Substitute the values]

\( =156^{\circ} \) [Divide]

Hence, each interior angle measures \( 156^{\circ} \).

Example 6: Find the measure of the unknown angle.

Solution:

The polygon has 6 sides. Let us first find the sum of its interior angles.

Sum of interior angles, \(S=(n-2)\times 180^{\circ}\) [Write the formula]

\(S=(6-2)\times 180^{\circ}\) [Substitute 6 for n]

\(S=(4)\times 180^{\circ}\) [Subtract]

\(S=720^{\circ}\) [Multiply]

As per the figure the interior angles of the polygon are \( 140^{\circ},~150^{\circ},~130^{\circ},~x \) and two right angles, which are \( 90^{\circ}, \) each. The sum of these angles can be written as:

\( 140^{\circ} + 150^{\circ} + 130^{\circ} + x + 90^{\circ} + 90^{\circ} = 720^{\circ} \)

\( 600^{\circ} + x = 720^{\circ} \) [Add]

\( x = 720^{\circ} – 600^{\circ} \) [Subtract 600 on both sides]

\( x = 80^{\circ} \)

Hence, the measure of unknown angle \( x\) is \( 80^{\circ} \).

Quadrilaterals are a type of polygon with 4 sides. Example: square, rectangle, rhombus, kite, trapezoid and a parallelogram.

The sum of interior angles can be obtained by dividing it into triangles. For example, the pentagon below has been divided into 3 triangles by joining the vertices 2 with 5 and 3 with 5.

Now the sum of the interior angles of the pentagon will be the sum of the interior angles of the three triangles, that is, \( 3\times 180^{\circ} = 540^{\circ} \).

Similarly we can divide other polygons into triangles and find the sum of their interior angles.

[Note: Sum of interior angles of a triangle is \( 180^{\circ} \)]

A trapezoid is a quadrilateral that is a polygon with 4 sides. Hence, the sum of the interior angles in a trapezoid will be \( 360^{\circ} \).

A polygon in which all interior angles measure the same is known as an equiangular polygon.

A polygon in which all sides measure the same in length is known as an equilateral polygon.

[Note: A regular polygon is both equiangular and equilateral.]