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ACT Math : How to find an angle in a polygon

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Example Questions

Example Question #601 : Plane Geometry

In the diagram below, what is $\theta$ equal to?

Hexagon_sides

Possible Answers:

$\sin^{-1}\left ( \frac{r}{s} \right )$

$\tan^{-1}\left ( \frac{r}{s} \right )$

$\tan^{-1}\left ( \frac{s}{r} \right )$

$\tan \left ( \frac{r}{s} \right )$

$\tan \left ( \frac{s}{r} \right )$

Correct answer:

$\tan^{-1}\left ( \frac{s}{r} \right )$

Explanation:

The figure given is a hexagon with an embedded triangle. The fact that it is embedded in a triangle is mainly to throw you off, as it has little to no consequence on the correct answer. Of the available answer choices, you must choose a relationship that would give the value of $\theta$ . Tangent describes the relationship between an angle and the opposite and adjacent sides of that angle. Or in other words, tan $\theta$ = opposite side/adjacent side. However, when solving for an angle, we must use the inverse function. Therefore, if we know the opposite and adjacent sides are, we can use the inverse of the tangent, or arctangent (tan^-1), of $\frac{s}{r}$ to find $\theta$ .

Thus,

$\theta =\tan^{-1}\left ( \frac{s}{r} \right )$

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Example Question #1 : Other Polygons

Shape_1

What is the value of angle in the figure above?

Possible Answers:

$110^{\circ}$

$75^{\circ}$

$50^{\circ}$

$35^{\circ}$

$100^{\circ}$

Correct answer:

$100^{\circ}$

Explanation:

Begin by noticing that the upper-right angle of this figure is supplementary to $70^{\circ}$ . This means that it is $180^{\circ}-70^{\circ}=110^{\circ}$ :

Shape_1

Now, a quadrilateral has a total of $360^{\circ}$ . This is computed by the formula degrees = 180 * (s-2) , where represents the number of sides. Thus, we know:

$110^{\circ}+70^{\circ}+80^{\circ}+x=360^{\circ}$

This is the same as $260^{\circ} + x = 360^{\circ}$

Solving for , we get:

$x=100^{\circ}$

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Example Question #603 : Plane Geometry

Fig_2

What is the angle measure for the largest unknown angle in the figure above? Round to the nearest hundredth.

Possible Answers:

$186.36^{\circ}$

$84.14^{\circ}$

$140.14^{\circ}$

$74.55^{\circ}$

$149.1^{\circ}$

Correct answer:

$186.36^{\circ}$

Explanation:

The total degree measure of a given figure is given by the equation 180*(s-2) , where represents the number of sides in the figure. For this figure, it is: $180*(6-2)=180*4 = 720^{\circ}$

Therefore, we know that the sum of the angles must equal $720^{\circ}$ . This gives us the equation:

$90^{\circ}+120^{\circ}+100^{\circ}+x+2.5x+2x=720^{\circ}$

Simplifying, this is:

$310^{\circ}+5.5x=720^{\circ}$

Now, just solve for :

5.5x=410

$x=\frac{410}{5.5}$

The largest of the unknown angles is 2.5x or $2.5 * \frac{410}{5.5}=186.36363636363636...^{\circ}$

Rounding, this is $186.36^{\circ}$ .

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Example Question #4 : How To Find An Angle In A Polygon

What is the interior angle of a $\textup{9-sided}$ polygon (a nonagon)? Round answer to the nearest hundredth if necessary.

Possible Answers:

$140^{\circ}$

$147.27^{\circ}$

$1260^{\circ}$

$128.57^{\circ}$

$360^{\circ}$

Correct answer:

$140^{\circ}$

Explanation:

To find the interior angle of an sided polygon, first find the total number of degrees in the polygon by the formula:
$(n-2)*180^{\circ}$ . For us that yields:
$(9-2)*180^{\circ} = 1260^{\circ}$ . Next we divide the total number of degrees by the number of sides:

$1260^{\circ}/9 = 140^{\circ}$

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Example Question #12 : Other Polygons

What is the total number of degrees in a $\textup{12-sided}$ polygon?

Possible Answers:

$2160^{\circ}$

$1800^{\circ}$

$144^{\circ}$

$150^{\circ}$

$1440^{\circ}$

Correct answer:

$1800^{\circ}$

Explanation:

To find the total number of degrees in an -sided polygon, use the formula:

(n-2)*180 thus we see that (12-2)*180 = $1800^{\circ}$

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Example Question #5 : How To Find An Angle In A Polygon

What is the interior angle of a $\textup{16-sided}$ polygon?

Possible Answers:

$360^{\circ}$

$2520^{\circ}$

$154.29^{\circ}$

$157.5^{\circ}$

$160^{\circ}$

Correct answer:

$157.5^{\circ}$

Explanation:

To find the interior angle of a regular, -sided polygon, use the formula:

((n-2)*180)/n :

Thus we see that $(16-2)*180^{\circ} = 2520^{\circ}$ and

$2520^{\circ}/16 = 157.5^{\circ}$

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Example Question #6 : How To Find An Angle In A Polygon

What is the total number of degrees in a $\textup{7-sided}$ polygon?

Possible Answers:

$540^{\circ}$

$360^{\circ}$

$900^{\circ}$

$128.57^{\circ}$

$1260^{\circ}$

Correct answer:

$900^{\circ}$

Explanation:

To find the total number of degrees in an -sided polygon, use the formula:

$(n-2)*180^{\circ}$ . Thus we see that:

$(7-2)*180 = 900^{\circ}$

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Example Question #611 : Plane Geometry

How many degrees are in the interior of an octagon (an 8-sided polygon)?

Possible Answers:

$1080^{\circ}$

$900^{\circ}$

$1440^{\circ}$

$1260^{\circ}$

$135^{\circ}$

Correct answer:

$1080^{\circ}$

Explanation:

To find the total number of degrees in an -sided polygon, use the formula:

$(n-2)*180^{\circ}$ . Thus, for an octagon we have:

$(8-2)*180^{\circ} = 1080^{\circ}$

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Example Question #612 : Plane Geometry

What is the interior angle of a pentagon?

Possible Answers:

$72^{\circ}$

$540^{\circ}$

$110^{\circ}$

$128.57^{\circ}$

$108^{\circ}$

Correct answer:

$108^{\circ}$

Explanation:

To find the interior angle of an -sided polygon, use the formula:

$((n-2)*180^{\circ})/n$
we see for our polygon this yields:

$((5-2)180^{\circ})/5=108^{\circ}$

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Example Question #9 : How To Find An Angle In A Polygon

What is the interior angle of a ten-sided polygon?

Possible Answers:

$1440^{\circ}$

$150^{\circ}$

$144^{\circ}$

$180^{\circ}$

$360^{\circ}$

Correct answer:

$144^{\circ}$

Explanation:

To find the interior angle of an -sided polygon, use the equation:

$((n-2)*180^{\circ})/n$

Plugging in 10 for yields:

$((10-2)*180^{\circ}/10)=144^{\circ}$

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ACT Math : How to find an angle in a polygon

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #601 : Plane Geometry

Example Question #1 : Other Polygons

Example Question #603 : Plane Geometry

Example Question #4 : How To Find An Angle In A Polygon

Example Question #12 : Other Polygons

Example Question #5 : How To Find An Angle In A Polygon

Example Question #6 : How To Find An Angle In A Polygon

Example Question #611 : Plane Geometry

Example Question #612 : Plane Geometry

Example Question #9 : How To Find An Angle In A Polygon

All ACT Math Resources

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