# If the measure of each interior angle of a regular polygon is 171, what is the number of sides in the polygon?

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28
Feb 18, 2016

The polygon will have 40 sides.

#### Explanation:

The measure of the interior angles of a polygon is determined by the formula

$\frac{\left(n - 2\right) 180}{n}$ = interior angle measure

were $n$ is the number of angles and sides of the polygon.

We know that the interior angles of the polygon in the question have measures of ${171}^{o}$

$\frac{\left(n - 2\right) 180}{n} = {171}^{o}$
$\left(\left(n - 2\right) 180\right) = {171}^{o} \left(n\right)$
$180 n - 360 = 171 n$
$- 360 = 171 n - 180 n$
$- 360 = - 9 n$
$- \frac{360}{-} 9 = n$
$40 = n$

The polygon will have 40 sides.

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May 17, 2018

Use the exterior angle to find there are $40$ sides

#### Explanation:

The interior angle and the exterior angle of a polygon are adjacent supplementary angles. This means that they add to 180°

Each exterior angle = 180°-171° = 9°

The sum of the exterior angles of any polygon is 360°

$\text{Number of sides" = (360°)/"exterior angle}$

"Number of sides" = (360°)/(9°) = 40 .

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