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5.27: Interior Angles in Convex Polygons

Last updated

Jun 15, 2022
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- 5.26: Determine Missing Angle Measures
- 5.28: Exterior Angles in Convex Polygons

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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.

f-d_389ff0723ea49e71f9ec7b5c21806d212be78a36ee4921973da5c040+IMAGE_TINY+IMAGE_TINY.png — Figure $\PageIndex{1}$

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 $(n−2)\times 180^{\circ}$ .

f-d_dc6113c85bfcaef0605b41b9ea2e6f06809d9fc3633e6525fcd7b83b+IMAGE_TINY+IMAGE_TINY.png — Figure $\PageIndex{2}$

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

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 $\dfrac{(n−2)\times 180^{\circ}}{n}$ .

f-d_c35906e4638039ad92c1150cea19cce1d19012c9780b4e444b8b1185+IMAGE_TINY+IMAGE_TINY.png — Figure $\PageIndex{3}$

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).

Figure $\PageIndex{5}$
Figure $\PageIndex{6}$
Figure $\PageIndex{7}$
Figure $\PageIndex{8}$
Figure $\PageIndex{9}$
Figure $\PageIndex{10}$
Figure $\PageIndex{11}$
Figure $\PageIndex{12}$
Figure $\PageIndex{13}$

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

Review

Review (Answers)

Resources

Vocabulary

Additional Resources

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