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SAT Math : How to find an angle in a polygon

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Example Question #372 : Geometry

To the nearest whole degree, give the measure of each interior angle of a regular polygon with 17 sides.

Possible Answers:

$174^{\circ }$ $\displaystyle 174^{\circ }$

$164^{\circ }$ $\displaystyle 164^{\circ }$

$154^{\circ }$ $\displaystyle 154^{\circ }$

$169^{\circ }$ $\displaystyle 169^{\circ }$

$159^{\circ }$ $\displaystyle 159^{\circ }$

Correct answer:

$159^{\circ }$ $\displaystyle 159^{\circ }$

Explanation:

The measure of each interior angle of an $\displaystyle N$-sided polygon can be calculated using the formula

$m = \frac{(N-2)180^{\circ }}{N}$ $\displaystyle m = \frac{(N-2)180^{\circ }}{N}$

Setting N = 17 $\displaystyle N = 17$:

$m = \frac{(17-2)180^{\circ }}{17} = \frac{(15)180^{\circ }}{17} = \frac{2,700^{\circ }}{17} = 158 \frac{14}{17}^{\circ }$ $\displaystyle m = \frac{(17-2)180^{\circ }}{17} = \frac{(15)180^{\circ }}{17} = \frac{2,700^{\circ }}{17} = 158 \frac{14}{17}^{\circ }$

The correct choice is therefore $159^{\circ }$ $\displaystyle 159^{\circ }$.

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Example Question #373 : Geometry

Each interior angle of a regular polygon has measure $175 ^{\circ }$ $\displaystyle 175 ^{\circ }$. How many sides does the polygon have?

Possible Answers:

108 $\displaystyle 108$

144 $\displaystyle 144$

$\displaystyle 72$

$\displaystyle 36$

$\displaystyle 18$

Correct answer:

$\displaystyle 72$

Explanation:

The easiest way to work this is arguably to examine the exterior angles, each of which forms a linear pair with an interior angle. If an interior angle measures $175 ^{\circ }$ $\displaystyle 175 ^{\circ }$, then each exterior angle, which is supplementary to an interior angle, measures

$180 ^{\circ }- 175 ^{\circ } = 5^{\circ }$ $\displaystyle 180 ^{\circ }- 175 ^{\circ } = 5^{\circ }$

The measures of the exterior angles of a polygon, one per vertex, total $360 ^{\circ }$ $\displaystyle 360 ^{\circ }$; in a regular polygon, they are congruent, so if there are $\displaystyle N$ such angles, each measures $\frac{360^{\circ }}{N}$ $\displaystyle \frac{360^{\circ }}{N}$. Since the number of vertices is equal to the number of sides, if we set this equal to $5^{\circ }$ $\displaystyle 5^{\circ }$ and solve for $\displaystyle N$, we will find the number of sides.

$\frac{360 }{N} = 5$ $\displaystyle \frac{360 }{N} = 5$

Multiply both sides by $\frac{N}{5}$ $\displaystyle \frac{N}{5}$:

$\frac{360 }{N}\cdot \frac{N}{5}= 5 \cdot \frac{N}{5}$ $\displaystyle \frac{360 }{N}\cdot \frac{N}{5}= 5 \cdot \frac{N}{5}$

$\frac{360 }{ 5}= N$ $\displaystyle \frac{360 }{ 5}= N$

72 = N $\displaystyle 72 = N$

The polygon has 72 vertices and, thus, 72 sides.

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Example Question #1 : How To Find An Angle In A Polygon

A regular polygon has a measure of $140^\circ$ $\displaystyle 140^\circ$ for each of its internal angles. How many sides does it have?

Possible Answers:

$\displaystyle 6$

$\displaystyle 7$

$\displaystyle 8$

$\displaystyle 9$

Correct answer:

$\displaystyle 9$

Explanation:

To determine the measure of the angles of a regular polygon use:

Angle = (n – 2) x 180° / n

Thus, (n – 2) x 180° / n = 140°

180° n - 360° = 140° n

40° n = 360°

n = 360° / 40° = 9

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Example Question #11 : How To Find An Angle In A Polygon

A regular seven sided polygon has a side length of 14”. What is the measurement of one of the interior angles of the polygon?

Possible Answers:

257.14 degrees

252 degrees

154.28 degrees

180 degrees

128.57 degrees

Correct answer:

128.57 degrees

Explanation:

The formula for of interior angles based on a polygon with a number of side n is:

Each Interior Angle = (n-2)*180/n

= (7-2)*180/7 = 128.57 degrees

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SAT Math : How to find an angle in a polygon

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #372 : Geometry

Example Question #373 : Geometry

Example Question #1 : How To Find An Angle In A Polygon

Example Question #11 : How To Find An Angle In A Polygon

All SAT Math Resources

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