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

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

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

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

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 \begin{align*}x^\circ, x^\circ, 2x^\circ, 2x^\circ,\end{align*}x,x,2x,2x, and \begin{align*}2x^\circ\end{align*}2x. What is \begin{align*}x\end{align*}x?

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

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

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

Example 2

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

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

Example 3

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

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

\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 \begin{align*}n = 9.\end{align*}

\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 \begin{align*}\frac{1260^\circ}{9} = 140^\circ\end{align*}.

Example 5

An interior angle in a regular polygon is \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 \begin{align*}135^\circ\end{align*} and solve for \begin{align*}n\end{align*}.

\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 \begin{align*}n-\end{align*}gon
3 \begin{align*}60^\circ\end{align*}
4 \begin{align*}360^\circ\end{align*}
5 \begin{align*}540^\circ\end{align*} \begin{align*}108^\circ\end{align*}
6 \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 \begin{align*}4320^\circ\end{align*}. How many sides does the polygon have?
  4. The sum of the interior angles of a polygon is \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 \begin{align*}156^\circ\end{align*}. How many sides does it have?
  8. Each interior angle in an equiangular polygon is \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 \begin{align*}x^\circ, (x + 1)^\circ, (x + 2)^\circ, (x + 3)^\circ, (x + 4)^\circ,\end{align*} and \begin{align*}(x + 5)^\circ.\end{align*} What is \begin{align*}x\end{align*}?

Review (Answers)

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

Resources

 

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Vocabulary

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.

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