# 5.27: Interior Angles in Convex Polygons

- Page ID
- 5012

Use the formula \((x - 2)180\) to find the sum of the interior angles of any polygon.

The **interior angle** of a polygon is one of the angles on the inside, as shown in the picture below. A polygon has the same number of interior angles as it does sides.

The sum of the interior angles in a polygon depends on the number of sides it has. The **Polygon Sum Formula** states that for any n−gon, the interior angles add up to \((n−2)\times 180^{\circ}\).

\(\begin{aligned}\rightarrow n=8 \\ (8−2) &\times 180^{\circ} \\ 6 &\times 180^{\circ} \\ &1080^{\circ}\end{aligned}\)

Once you know the sum of the interior angles in a polygon it is easy to find the measure of ONE interior angle if the polygon is **regular**: all sides are congruent and all angles are congruent. Just divide the sum of the angles by the number of sides.

**Regular Polygon Interior Angle Formula:** For any __equiangular__ n−gon, the measure of each angle is \(\dfrac{(n−2)\times 180^{\circ}}{n}\).

In the picture below, if all eight angles are congruent then each angle is \(\dfrac{(8−2)\times 180^{\circ}}{8}=\dfrac{6\times 180^{\circ}}{8}=\dfrac{1080^{\circ}}{8}=135^{\circ}\).

What if you were given an equiangular seven-sided convex polygon? How could you determine the measure of its interior angles?

Example \(\PageIndex{1}\)

The interior angles of a pentagon are \(x^{\circ}\), \(x^{\circ}\), \(2x^{\circ}\), \(2x^{\circ}\), and \(2x^{\circ}\). What is \(x\)?

**Solution**

From the Polygon Sum Formula we know that a pentagon has interior angles that sum to \((5−2)\times 180^{\circ}=540^{\circ}\).

Write an equation and solve for x.

\(\begin{aligned} x^{\circ}+x^{\circ}+2x^{\circ}+2x^{\circ}+2x^{\circ}&=540^{\circ} \\ 8x&=540 \\ x&=67.5\end{aligned}\)

Example \(\PageIndex{2}\)

What is the sum of the interior angles in a 100-gon?

**Solution**

Use the Polygon Sum Formula. \((100−2)\times 180^{\circ}=17,640^{\circ}\).

Example \(\PageIndex{3}\)

The interior angles of a polygon add up to \(1980^{\circ}\). How many sides does it have?

**Solution**

Use the Polygon Sum Formula and solve for n\).

\(\begin{aligned} (n−2)\times 180^{\circ}&=1980^{\circ} \\ 180^{\circ}n−360^{\circ}&=1980^{\circ} \\ 180^{\circ}n&=2340^{\circ} \\ n&=13\end{aligned}\)

The polygon has 13 sides.

Example \(\PageIndex{4}\)

How many degrees does **each angle** in an

__equiangular__nonagon have?

**Solution**

First we need to find the sum of the interior angles; set \(n=9\).

\((9−2)\times 180^{\circ}=7\times 180^{\circ}=1260^{\circ}\)

“Equiangular” tells us every angle is equal. So, each angle is \(\dfrac{1260^{\circ}}{9}=140^{\circ}\).

Example \(\PageIndex{5}\)

An interior angle in a regular polygon is \(135^{\circ}\). How many sides does this polygon have?

**Solution**

Here, we will set the Regular Polygon Interior Angle Formula equal to \(135^{\circ}\) and solve for n.

\(\begin{aligned} \dfrac{(n−2)\times 180^{\circ}}{n}&=135^{\circ} \\ 180^{\circ}n−360^{\circ}−360^{\circ}&=135^{\circ}n \\ n&=−45^{\circ} \\ n&=8\qquad \text{The polygon is an octagon.} \end{aligned}\)

## Review

- Fill in the table.

# of sides |
Sum of the Interior Angles |
Measure of Each Interior Angle in a Regular n−gon |
---|---|---|

3 | \(60^{\circ}\) | |

4 | \(360^{\circ}\) | |

5 | \(540^{\circ}\) | \(108^{\circ}\) |

6 | \(120^{\circ}\) | |

7 | ||

8 | ||

9 | ||

10 | ||

11 | ||

12 |

- What is the sum of the angles in a 15-gon?
- What is the sum of the angles in a 23-gon?
- The sum of the interior angles of a polygon is \(4320^{\circ}\). How many sides does the polygon have?
- The sum of the interior angles of a polygon is \(3240^{\circ}\). How many sides does the polygon have?
- What is the measure of each angle in a regular 16-gon?
- What is the measure of each angle in an equiangular 24-gon?
- Each interior angle in a regular polygon is \(156^{\circ}\). How many sides does it have?
- Each interior angle in an equiangular polygon is \(90^{\circ}\). How many sides does it have?

For questions 10-18, find the value of the missing variable(s).

- The interior angles of a hexagon are \(x^{\circ}\), \((x+1)^{\circ}\), \((x+2)^{\circ}\), \((x+3)^{\circ}\),\((x+4)^{\circ}\), and \((x+5)^{\circ}\). What is \(x\)?

## Review (Answers)

To see the Review answers, open this PDF file and look for section 6.1.

## Resources

## Vocabulary

Term | Definition |
---|---|

Interior angles |
Interior angles are the angles inside a figure. |

Polygon Sum Formula |
The Polygon Sum Formula states that for any polygon with n sides, the interior angles add up to \((n−2)\times 180\) degrees. |

## Additional Resources

Interactive Element

Video: Interior and Exterior Angles of a Polygon

Activities: Interior Angles in Convex Polygons Discussion Questions

Study Aids: Polygons Study Guide

Practice: Interior Angles in Convex Polygons

Real World: Interior Angles In Convex Polygons