**Middle School Number Sense Lesson 49: Finding the Interior Angle of a Polygon**

Today we will continue our discussion from last week about polygons and cover

**interior angles**. (You may want to review

**AngleExterior**before continuing). This concept appeared

**4 times**last year, with a median placement at

**question # 49**.

**Number Dojo Level: 211**

We have already established that:

- The sum of the exterior angles of any polygon is 360°, and
- At any vertex, the exterior angle and the interior angle are
**supplementary**(their measures add up to 180°).

**How to solve:**

To find the

**measure of one interior angle**, we can first find the exterior angle and subtract it from 180°. Or we can use the formula

**180(n - 2)/n**, where

**n**is the number of sides of the polygon.

To find the

**sum of the measures of the interior angles**of a polygon, we can use the formula

**180(n - 2)**because we are multiplying

**180(n - 2)/n**by

**n**.

**Example 1: The measure of an interior angle of a regular hexagon is ___°**

- A hexagon has 6 sides, so its exterior angle measures 360/6 or
**60°**. - The interior angle is (180 - 60) =
**120°**.

**Example 2: The measure of an interior angle of a regular decagon is ___°**

- A decagon has 10 sides, so its exterior angle measures 360/10 or
**36°**. - The interior angle is (180 - 36) =
**144°**.

**Example 3: The interior angle of a regular icosagon measures ___°**

- An icosagon has 20 sides, so its exterior angle measures 360/20 or
**18**°. - The interior angles is (180 - 18) =
**162**°.

**Example 4: The sum of the interior angles of a regular octagon is ___°**

- An octagon has 8 sides.
- 180(n - 2) = 180(8 - 2) = 180(6) =
**1080°**.

**Example 5: The sum of the measures of the interior angles of a 20-sided polygon is ___°**

- 180(20 - 2) = 180(18) =
**3240°**.

**Example 6: The number of sides of a regular polygon with an interior angle measure of 156° is ___**

- If the interior angle is 156°, the exterior angle is (180 - 156) =
**24**°. - n = 360/24 =
**15**.

**Here's a free worksheet to help you practice AngleInterior:**

angleinterior.pdf |

**Up Next for Middle School: EstMultDeci**