How to find an angle in a polygon - High School Math

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Question

What is the magnitude of the interior angle of a regular nonagon?

Answer

The equation to calculate the magnitude of an interior angle is $\frac{180(n-2)}{n}=int\ angle$ , where is equal to the number of sides.

For our question, .

$int\ angle= \frac{180(9-2)}{9}=\frac{(180)(7)}{9}=140^o$

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Question

What is the interior angle measure of any regular heptagon?

Answer

To find the angle of any regular polygon you find the number of sides, . In this example, .

You then subtract 2 from the number of sides yielding 5.

Take 5 and multiply it by 180 degrees to yield the total number of degrees in the regular heptagon.

Then to find one individual angle we divide 900 by the total number of angles, 7.

$\frac{900}{7}=128.6$

The answer is .

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Question

A regular polygon with sides has exterior angles that measure $1.5^{\circ }$ each. How many sides does the polygon have?

Answer

The sum of the exterior angles of any polygon, one per vertex, is $360 ^{\circ }$ . As each angle measures $1.5^{\circ }$ , just divide 360 by 1.5 to get the number of angles.

$360 \div 1.5=240$

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Question

What is the interior angle measure of any regular nonagon?

Answer

To find the angle of any regular polygon you find the number of sides , which in this example is .

You then subtract from the number of sides yielding .

Take and multiply it by degrees to yield a total number of degrees in the regular nonagon.

Then to find one individual angle we divide by the total number of angles .

$\frac{1260}{9}=140$

The answer is .

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Question

What is the measure of one exterior angle of a regular seventeen-sided polygon (nearest tenth of a degree)?

Answer

The sum of the measures of the exterior angles of any polygon, one per vertex, is $360^{\circ}$ . In a regular polygon, all of these angles are congruent, so divide 360 by 17 to get the measure of one exterior angle:

$360 \div 17 \approx 21.2^{\circ }$

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Question

What is the measure of one exterior angle of a regular twenty-three-sided polygon (nearest tenth of a degree)?

Answer

The sum of the measures of the exterior angles of any polygon, one per vertex, is $360^{\circ}$ . In a regular polygon, all of these angles are congruent, so divide 360 by 23 to get the measure of one exterior angle:

$360 \div 23 \approx 15.7 ^{\circ }$

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Question

What is the measure of one interior angle of a regular twenty-three-sided polygon (nearest tenth of a degree)?

Answer

The measure of each interior angle of a regular polygon with sides is $\frac{180 (N-2)}{N}$ . We can substitute to obtain the angle measure:

$\frac{180 (N-2)}{N} = \frac{180 (23-2)}{23} = \frac{180 \cdot 21}{23}\approx 164.3^{\circ }$

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Question

A regular polygon has interior angles which measure $162^{\circ }$ each. How many sides does the polygon have?

Answer

The easiest way to answer this is to note that, since an interior angle and an exterior angle form a linear pair - and thus, a supplementary pair - each exterior angle would have measure $180-162 = 18^{\circ }$ . Since 360 divided by the number of sides of a regular polygon is equal to the measure of one of its exterior angles, we are seeking such that

$\frac{360}{N} = 18$

Solve for :

$\frac{360}{N}\cdot N= 18\cdot N$

$18N \div 18= 360\div 18$

The polygon has 20 sides.

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