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ACT Math : How to find an angle in a hexagon

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

The sum of all the angles inside of a regular hexagon is $720^{\circ}$ $\displaystyle 720^{\circ}$ . Determine the value of one angle.

Possible Answers:

$220^{\circ}$ $\displaystyle 220^{\circ}$

$180^{\circ}$ $\displaystyle 180^{\circ}$

$120^{\circ}$ $\displaystyle 120^{\circ}$

$160^{\circ}$ $\displaystyle 160^{\circ}$

$140^{\circ}$ $\displaystyle 140^{\circ}$

Correct answer:

$120^{\circ}$ $\displaystyle 120^{\circ}$

Explanation:

In a regular hexagon, all of the sides are the same length, and all of the angles are equivalent. The problem tells us that all of the angles inside the hexagon sum to $720^{\circ}$ $\displaystyle 720^{\circ}$ . To find the value of one angle, we must divide $720^{\circ}$ $\displaystyle 720^{\circ}$ by $\displaystyle 6$ , since there are $\displaystyle 6$ angles inside of a hexagon.

$\frac{720 ^{\circ}}{6}=120^{\circ}$ $\displaystyle \frac{720 ^{\circ}}{6}=120^{\circ}$

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Example Question #2 : How To Find An Angle In A Hexagon

Hexex11

All of the angles marked are exterior angles.

What is the value of $\displaystyle a$ in degrees? Round to the nearest hundredth.

Possible Answers:

$25.20^{\circ}$ $\displaystyle 25.20^{\circ}$

$32.73^{\circ}$ $\displaystyle 32.73^{\circ}$

$40.00^{\circ}$ $\displaystyle 40.00^{\circ}$

$65.45^{\circ}$ $\displaystyle 65.45^{\circ}$

$31.14^{\circ}$ $\displaystyle 31.14^{\circ}$

Correct answer:

$32.73^{\circ}$ $\displaystyle 32.73^{\circ}$

Explanation:

There are two key things for a question like this. The first is to know that a polygon has a total degree measure of:

$\displaystyle 180*(s-2)$ , where $\displaystyle s$ is the number of sides.

Therefore, a hexagon like this one has:

$180*4=720^{\circ}$ $\displaystyle 180*4=720^{\circ}$ .

Next, you should remember that all of the exterior angles listed are supplementary to their correlative interior angles. This lets you draw the following figure:

Hexex12

Now, you just have to manage your algebra well. You must sum up all of the interior angles and set them equal to 720 $\displaystyle 720$ . Since there are $\displaystyle 6$ angles, you know that the numeric portion will be 180*6 $\displaystyle 180*6$ or 1080 $\displaystyle 1080$ . Thus, you can write:

$\displaystyle 1080-a-2a-a-3a-a-3a=720$

Simplify and solve for $\displaystyle a$ :

$\displaystyle 1080-11a=720$

$\displaystyle -11a=-360$

$a=\frac{360}{11}$ $\displaystyle a=\frac{360}{11}$

This is $\displaystyle 32.72727272727272...$ or $32.73^{\circ}$ $\displaystyle 32.73^{\circ}$ .

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Example Question #2 : How To Find An Angle In A Hexagon

Hexex21

The figure above is a hexagon. All of the angles listed (except the interior one) are exterior angles to the hexagon's interior angles.

What is the value of $\displaystyle x$ ?

Possible Answers:

$60^{\circ}$ $\displaystyle 60^{\circ}$

$120^{\circ}$ $\displaystyle 120^{\circ}$

$75^{\circ}$ $\displaystyle 75^{\circ}$

$150^{\circ}$ $\displaystyle 150^{\circ}$

$50^{\circ}$ $\displaystyle 50^{\circ}$

Correct answer:

$50^{\circ}$ $\displaystyle 50^{\circ}$

Explanation:

There are two key things for a question like this. The first is to know that a polygon has a total degree measure of:

$\displaystyle 180*(s-2)$ , where $\displaystyle s$ is the number of sides.

Therefore, a hexagon like this one has:

$180*4=720^{\circ}$ $\displaystyle 180*4=720^{\circ}$ .

Next, you should remember that all of the exterior angles listed are supplementary to their correlative interior angles. This lets you draw the following figure:

Hexex22

Now, you just have to manage your algebra well. You must sum up all of the interior angles and set them equal to 720 $\displaystyle 720$ . Thus, you can write:

$\displaystyle 140+160+150+85+135+x=720$

Solve for $\displaystyle x$ :

$\displaystyle 670+x=720$

$x=50^{\circ}$ $\displaystyle x=50^{\circ}$

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ACT Math : How to find an angle in a hexagon

Study concepts, example questions & explanations for ACT Math

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

Example Question #2 : How To Find An Angle In A Hexagon

Example Question #2 : How To Find An Angle In A Hexagon

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