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# Interior Angles in Convex Polygons

## Angles inside a closed figure with straight sides.

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Interior Angles in Convex Polygons

### Interior Angles in Convex Polygons

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\begin{align*}n-\end{align*}gon, the interior angles add up to (n2)×180.\begin{align*}(n - 2) \times 180^\circ.\end{align*}

n=8(82)61×180×180080\begin{align*}\rightarrow n = 8& \\ (8 - 2) & \times 180^\circ\\ 6 & \times 180^\circ\\ 1& 080^\circ\end{align*}

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\begin{align*}n-\end{align*}gon, the measure of each angle is (n2)×180n\begin{align*}\frac{(n-2) \times 180^\circ}{n}\end{align*}.

In the picture below, if all eight angles are congruent then each angle is (82)×1808=6×1808=10808=135\begin{align*}\frac{(8 - 2) \times 180^\circ}{8} = \frac{6 \times 180^\circ}{8} = \frac{1080^\circ}{8} = 135^\circ\end{align*}.

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

### Examples

#### Example 1

The interior angles of a pentagon are x,x,2x,2x,\begin{align*}x^\circ, x^\circ, 2x^\circ, 2x^\circ,\end{align*} and 2x\begin{align*}2x^\circ\end{align*}. What is x\begin{align*}x\end{align*}?

From the Polygon Sum Formula we know that a pentagon has interior angles that sum to (52)×180=540\begin{align*}(5-2) \times 180^\circ=540^\circ\end{align*}.

Write an equation and solve for x\begin{align*}x\end{align*}.

x+x+2x+2x+2x8xx=540=540=67.5\begin{align*} x^\circ + x^\circ + 2x^\circ + 2x^\circ + 2x^\circ&=540^\circ\\ 8x&=540\\x&=67.5\end{align*}

#### Example 2

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

Use the Polygon Sum Formula. (1002)×180=17,640\begin{align*}(100-2) \times 180^\circ=17,640^\circ\end{align*}.

#### Example 3

The interior angles of a polygon add up to 1980\begin{align*}1980^\circ\end{align*}. How many sides does it have?

Use the Polygon Sum Formula and solve for n\begin{align*}n\end{align*}.

(n2)×180180n360180nn=1980=1980=2340=13\begin{align*}(n - 2) \times 180^\circ & = 1980^\circ\\ 180^\circ n - 360^\circ & = 1980^\circ\\ 180^\circ n & = 2340^\circ\\ n & = 13\end{align*}

The polygon has 13 sides.

#### Example 4

How many degrees does each angle in an equiangular nonagon have?

First we need to find the sum of the interior angles; set n=9.\begin{align*}n = 9.\end{align*}

(92)×180=7×180=1260\begin{align*}(9 - 2) \times 180^\circ = 7 \times 180^\circ = 1260^\circ\end{align*}

“Equiangular” tells us every angle is equal. So, each angle is 12609=140\begin{align*}\frac{1260^\circ}{9} = 140^\circ\end{align*}.

#### Example 5

An interior angle in a regular polygon is 135\begin{align*}135^\circ\end{align*}. How many sides does this polygon have?

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

(n2)×180n180n360360n=135=135n=45n=8The polygon is an octagon.\begin{align*}\frac{(n - 2) \times 180^\circ}{n} & = 135^\circ\\ 180^\circ n - 360^\circ & = 135^\circ n\\ -360^\circ & = -45^\circ n\\ n & = 8 \qquad \quad \text{The polygon is an octagon}.\end{align*}

### Review

1. Fill in the table.
# of sides Sum of the Interior Angles Measure of Each Interior Angle in a Regular n\begin{align*}n-\end{align*}gon
3 60\begin{align*}60^\circ\end{align*}
4 360\begin{align*}360^\circ\end{align*}
5 540\begin{align*}540^\circ\end{align*} 108\begin{align*}108^\circ\end{align*}
6 120\begin{align*}120^\circ\end{align*}
7
8
9
10
11
12
1. What is the sum of the angles in a 15-gon?
2. What is the sum of the angles in a 23-gon?
3. The sum of the interior angles of a polygon is 4320\begin{align*}4320^\circ\end{align*}. How many sides does the polygon have?
4. The sum of the interior angles of a polygon is 3240\begin{align*}3240^\circ\end{align*}. How many sides does the polygon have?
5. What is the measure of each angle in a regular 16-gon?
6. What is the measure of each angle in an equiangular 24-gon?
7. Each interior angle in a regular polygon is 156\begin{align*}156^\circ\end{align*}. How many sides does it have?
8. Each interior angle in an equiangular polygon is 90\begin{align*}90^\circ\end{align*}. How many sides does it have?

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

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

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### Vocabulary Language: English

TermDefinition
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.