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Intermediate Geometry : Hexagons

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Example Questions

Example Question #111 : Hexagons

In terms of $\displaystyle x$, find the area of the regular hexagon with side lengths $\displaystyle 8x$.

Possible Answers:

$96x\sqrt3$ $\displaystyle 96x\sqrt3$

$64x^2\sqrt3$ $\displaystyle 64x^2\sqrt3$

$192x^2\sqrt3$ $\displaystyle 192x^2\sqrt3$

$96x^2\sqrt3$ $\displaystyle 96x^2\sqrt3$

Correct answer:

$96x^2\sqrt3$ $\displaystyle 96x^2\sqrt3$

Explanation:

Use the following formula to find the area of a regular hexagon:

$\text{Area}=\frac{3\sqrt3}{2}side^2$ $\displaystyle \text{Area}=\frac{3\sqrt3}{2}side^2$

Now, substitute in the value for the side length.

$\text{Area}=\frac{3\sqrt3}{2}(8x)^2=\frac{3\sqrt3}{2}(64x^2)=\frac{192x^2\sqrt3}{2}=96x^2\sqrt3$ $\displaystyle \text{Area}=\frac{3\sqrt3}{2}(8x)^2=\frac{3\sqrt3}{2}(64x^2)=\frac{192x^2\sqrt3}{2}=96x^2\sqrt3$

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Example Question #12 : How To Find The Area Of A Hexagon

Find the area of a regular hexagon with a side length of $\frac{1}{2}$ $\displaystyle \frac{1}{2}$.

Possible Answers:

$\frac{3\sqrt3}{6}$ $\displaystyle \frac{3\sqrt3}{6}$

$\frac{3\sqrt3}{4}$ $\displaystyle \frac{3\sqrt3}{4}$

$3\sqrt3$ $\displaystyle 3\sqrt3$

$\frac{3\sqrt3}{8}$ $\displaystyle \frac{3\sqrt3}{8}$

Correct answer:

$\frac{3\sqrt3}{8}$ $\displaystyle \frac{3\sqrt3}{8}$

Explanation:

Use the following formula to find the area of a regular hexagon:

$\text{Area}=\frac{3\sqrt3}{2}side^2$ $\displaystyle \text{Area}=\frac{3\sqrt3}{2}side^2$

Now, substitute in the value for the side length.

$\text{Area}=\frac{3\sqrt3}{2}\left(\frac{1}{2}\right)^2=\frac{3\sqrt3}{2}\left(\frac{1}{4}\right)=\frac{3\sqrt3}{8}$ $\displaystyle \text{Area}=\frac{3\sqrt3}{2}\left(\frac{1}{2}\right)^2=\frac{3\sqrt3}{2}\left(\frac{1}{4}\right)=\frac{3\sqrt3}{8}$

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Example Question #112 : Hexagons

Find the area of a regular hexagon with side lengths of $\frac{1}{3}$ $\displaystyle \frac{1}{3}$.

Possible Answers:

$\frac{\sqrt3}{2}$ $\displaystyle \frac{\sqrt3}{2}$

$\frac{\sqrt3}{4}$ $\displaystyle \frac{\sqrt3}{4}$

$\frac{\sqrt3}{6}$ $\displaystyle \frac{\sqrt3}{6}$

$\frac{\sqrt3}{8}$ $\displaystyle \frac{\sqrt3}{8}$

Correct answer:

$\frac{\sqrt3}{6}$ $\displaystyle \frac{\sqrt3}{6}$

Explanation:

Use the following formula to find the area of a regular hexagon:

$\text{Area}=\frac{3\sqrt3}{2}side^2$ $\displaystyle \text{Area}=\frac{3\sqrt3}{2}side^2$

Now, substitute in the value for the side length.

$\text{Area}=\frac{3\sqrt3}{2}\left(\frac{1}{3}\right)^2=\frac{3\sqrt3}{2}\left(\frac{1}{9}\right)=\frac{3\sqrt3}{18}=\frac{\sqrt3}{6}$ $\displaystyle \text{Area}=\frac{3\sqrt3}{2}\left(\frac{1}{3}\right)^2=\frac{3\sqrt3}{2}\left(\frac{1}{9}\right)=\frac{3\sqrt3}{18}=\frac{\sqrt3}{6}$

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Example Question #17 : How To Find The Area Of A Hexagon

A hexagon can be divided into six congruent equilateral triangles. The triangles have a sides that are 10 inches long. What is the area of the hexagon?

Possible Answers:

$150\sqrt 3 \; in^2$ $\displaystyle 150\sqrt 3 \; in^2$

$125\sqrt 3 \; in^2$ $\displaystyle 125\sqrt 3 \; in^2$

$300\sqrt 3 \; in^2$ $\displaystyle 300\sqrt 3 \; in^2$

$108\; in^2$ $\displaystyle 108\; in^2$

$75\; in^2$ $\displaystyle 75\; in^2$

Correct answer:

$150\sqrt 3 \; in^2$ $\displaystyle 150\sqrt 3 \; in^2$

Explanation:

To find the area of the hexagon we need to find the area of one of the equilateral triangles then multiply by six. To do this we have to find the height of one of the triangles. We find the height by using the Pythagorean theorem:

$\displaystyle a^2 + b^2 = c^2$

For one of our six equilateral triangles this looks like:

$\displaystyle 5^2 + b^2 = 10^2$

where 5 is half of one of our sides and b is the height of of the triangle (remember in an equilateral triangle the height will bisect the opposite side). We can now solve the equation for the missing hieght, b.

$25 + b^2 = 100 \rightarrow b^2 = 75 \text \; \rightarrow b = \sqrt 75 \rightarrow \newline b= \sqrt25\sqrt 3 \rightarrow b= 5\sqrt3$ $\displaystyle 25 + b^2 = 100 \rightarrow b^2 = 75 \text \; \rightarrow b = \sqrt 75 \rightarrow \newline b= \sqrt25\sqrt 3 \rightarrow b= 5\sqrt3$

So the height of one of our triangles is 5 times the square root of three. Now we have all the information we need. Using the triangle area formula, base times height divided by two we get:

$\frac{1}{2}(b x h) = \frac{1}{2}(10\times 5\sqrt3 )= \frac{1}{2}(50\sqrt3 ) = 25\sqrt3 \; in^2$ $\displaystyle \frac{1}{2}(b x h) = \frac{1}{2}(10\times 5\sqrt3 )= \frac{1}{2}(50\sqrt3 ) = 25\sqrt3 \; in^2$

This is the area for ONE of our triangles, but we have six! So we will just multiply this area by six:

$25\sqrt3 \; in^2 \times 6 = 150\sqrt3 \; in^2$ $\displaystyle 25\sqrt3 \; in^2 \times 6 = 150\sqrt3 \; in^2$

This is our final answer.

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Example Question #15 : How To Find The Area Of A Hexagon

A circle is placed inside a regular hexagon as shown in the figure.

If the radius of the circle is $\frac{2}{3}$ $\displaystyle \frac{2}{3}$, then find the area of the shaded region.

Possible Answers:

40.17 $\displaystyle 40.17$

47.81 $\displaystyle 47.81$

42.66 $\displaystyle 42.66$

38.39 $\displaystyle 38.39$

Correct answer:

40.17 $\displaystyle 40.17$

Explanation:

In order to find the area of the shaded region, we must first find the areas of the hexagon and the circle.

Recall how to find the area of a regular hexagon.

$\text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(\text{side})^2$ $\displaystyle \text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(\text{side})^2$

Substitute in the length of the given side to find the area of the hexagon.

$\text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(4)^2$ $\displaystyle \text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(4)^2$

$\text{Area of Regular Hexagon}=24\sqrt3$ $\displaystyle \text{Area of Regular Hexagon}=24\sqrt3$

Next, recall how to find the area of a circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$ $\displaystyle \text{Area of Circle}=\pi\times\text{radius}^2$

Substitute in the given radius to find the area of the circle.

$\text{Area of Circle}=\pi\times \left(\frac{2}{3}\right)^2$ $\displaystyle \text{Area of Circle}=\pi\times \left(\frac{2}{3}\right)^2$

$\text{Area of Circle}=\frac{4}{9}\pi$ $\displaystyle \text{Area of Circle}=\frac{4}{9}\pi$

Now, find the area of the shaded region by subtracting the area of the circle from the area of the hexagon.

$\text{Area of Shaded Region}=\text{Area of Hexagon}-\text{Area of Circle}$ $\displaystyle \text{Area of Shaded Region}=\text{Area of Hexagon}-\text{Area of Circle}$

$\text{Area of Shaded Region}=24\sqrt3-\frac{4}{9}\pi$ $\displaystyle \text{Area of Shaded Region}=24\sqrt3-\frac{4}{9}\pi$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=40.17$ $\displaystyle \text{Area of Shaded Region}=40.17$

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Example Question #924 : Plane Geometry

A circle is placed inside a regular hexagon as shown in the figure.

If the radius of the circle is $\displaystyle 2$, then find the area of the shaded region.

Possible Answers:

80.96 $\displaystyle 80.96$

77.69 $\displaystyle 77.69$

86.30 $\displaystyle 86.30$

78.21 $\displaystyle 78.21$

Correct answer:

80.96 $\displaystyle 80.96$

Explanation:

In order to find the area of the shaded region, we must first find the areas of the hexagon and the circle.

Recall how to find the area of a regular hexagon.

$\text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(\text{side})^2$ $\displaystyle \text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(\text{side})^2$

Substitute in the length of the given side to find the area of the hexagon.

$\text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(6)^2$ $\displaystyle \text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(6)^2$

$\text{Area of Regular Hexagon}=54\sqrt3$ $\displaystyle \text{Area of Regular Hexagon}=54\sqrt3$

Next, recall how to find the area of a circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$ $\displaystyle \text{Area of Circle}=\pi\times\text{radius}^2$

Substitute in the given radius to find the area of the circle.

$\text{Area of Circle}=\pi\times (2)^2$ $\displaystyle \text{Area of Circle}=\pi\times (2)^2$

$\text{Area of Circle}=4\pi$ $\displaystyle \text{Area of Circle}=4\pi$

Now, find the area of the shaded region by subtracting the area of the circle from the area of the hexagon.

$\text{Area of Shaded Region}=\text{Area of Hexagon}-\text{Area of Circle}$ $\displaystyle \text{Area of Shaded Region}=\text{Area of Hexagon}-\text{Area of Circle}$

$\text{Area of Shaded Region}=54\sqrt3-4\pi$ $\displaystyle \text{Area of Shaded Region}=54\sqrt3-4\pi$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=80.96$ $\displaystyle \text{Area of Shaded Region}=80.96$

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Example Question #17 : How To Find The Area Of A Hexagon

A circle is placed in a regular hexagon as shown in the figure below.

If the radius of the circle is $\displaystyle 4$, then find the area of the shaded region.

Possible Answers:

152.09 $\displaystyle 152.09$

172.51 $\displaystyle 172.51$

156.88 $\displaystyle 156.88$

160.18 $\displaystyle 160.18$

Correct answer:

160.18 $\displaystyle 160.18$

Explanation:

In order to find the area of the shaded region, we must first find the areas of the hexagon and the circle.

Recall how to find the area of a regular hexagon.

$\text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(\text{side})^2$ $\displaystyle \text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(\text{side})^2$

Substitute in the length of the given side to find the area of the hexagon.

$\text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(9)^2$ $\displaystyle \text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(9)^2$

$\text{Area of Regular Hexagon}=\frac{243\sqrt3}{2}$ $\displaystyle \text{Area of Regular Hexagon}=\frac{243\sqrt3}{2}$

Next, recall how to find the area of a circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$ $\displaystyle \text{Area of Circle}=\pi\times\text{radius}^2$

Substitute in the given radius to find the area of the circle.

$\text{Area of Circle}=\pi\times (4)^2$ $\displaystyle \text{Area of Circle}=\pi\times (4)^2$

$\text{Area of Circle}=16\pi$ $\displaystyle \text{Area of Circle}=16\pi$

Now, find the area of the shaded region by subtracting the area of the circle from the area of the hexagon.

$\text{Area of Shaded Region}=\text{Area of Hexagon}-\text{Area of Circle}$ $\displaystyle \text{Area of Shaded Region}=\text{Area of Hexagon}-\text{Area of Circle}$

$\text{Area of Shaded Region}=\frac{243\sqrt3}{2}-16\pi$ $\displaystyle \text{Area of Shaded Region}=\frac{243\sqrt3}{2}-16\pi$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=160.18$ $\displaystyle \text{Area of Shaded Region}=160.18$

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Example Question #115 : Hexagons

A circle is placed in a regular hexagon as shown in the figure.

If the radius of the circle is $\displaystyle 5$, then find the area of the shaded region.

Possible Answers:

253.98 $\displaystyle 253.98$

260.38 $\displaystyle 260.38$

221.02 $\displaystyle 221.02$

295.58 $\displaystyle 295.58$

Correct answer:

295.58 $\displaystyle 295.58$

Explanation:

In order to find the area of the shaded region, we must first find the areas of the hexagon and the circle.

Recall how to find the area of a regular hexagon.

$\text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(\text{side})^2$ $\displaystyle \text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(\text{side})^2$

Substitute in the length of the given side to find the area of the hexagon.

$\text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(12)^2$ $\displaystyle \text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(12)^2$

$\text{Area of Regular Hexagon}=216\sqrt3$ $\displaystyle \text{Area of Regular Hexagon}=216\sqrt3$

Next, recall how to find the area of a circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$ $\displaystyle \text{Area of Circle}=\pi\times\text{radius}^2$

Substitute in the given radius to find the area of the circle.

$\text{Area of Circle}=\pi\times (5)^2$ $\displaystyle \text{Area of Circle}=\pi\times (5)^2$

$\text{Area of Circle}=25\pi$ $\displaystyle \text{Area of Circle}=25\pi$

Now, find the area of the shaded region by subtracting the area of the circle from the area of the hexagon.

$\text{Area of Shaded Region}=\text{Area of Hexagon}-\text{Area of Circle}$ $\displaystyle \text{Area of Shaded Region}=\text{Area of Hexagon}-\text{Area of Circle}$

$\text{Area of Shaded Region}=216\sqrt3-25\pi$ $\displaystyle \text{Area of Shaded Region}=216\sqrt3-25\pi$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=295.58$ $\displaystyle \text{Area of Shaded Region}=295.58$

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Example Question #116 : Hexagons

A circle is placed in a regular hexagon as shown in the figure.

If the radius of the circle is $\displaystyle 3$, then find the area of the shaded region.

Possible Answers:

135.07 $\displaystyle 135.07$

142.06 $\displaystyle 142.06$

125.27 $\displaystyle 125.27$

138.00 $\displaystyle 138.00$

Correct answer:

138.00 $\displaystyle 138.00$

Explanation:

In order to find the area of the shaded region, we must first find the areas of the hexagon and the circle.

Recall how to find the area of a regular hexagon.

$\text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(\text{side})^2$ $\displaystyle \text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(\text{side})^2$

Substitute in the length of the given side to find the area of the hexagon.

$\text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(8)^2$ $\displaystyle \text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(8)^2$

$\text{Area of Regular Hexagon}=96\sqrt3$ $\displaystyle \text{Area of Regular Hexagon}=96\sqrt3$

Next, recall how to find the area of a circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$ $\displaystyle \text{Area of Circle}=\pi\times\text{radius}^2$

Substitute in the given radius to find the area of the circle.

$\text{Area of Circle}=\pi\times (3)^2$ $\displaystyle \text{Area of Circle}=\pi\times (3)^2$

$\text{Area of Circle}=9\pi$ $\displaystyle \text{Area of Circle}=9\pi$

Now, find the area of the shaded region by subtracting the area of the circle from the area of the hexagon.

$\text{Area of Shaded Region}=\text{Area of Hexagon}-\text{Area of Circle}$ $\displaystyle \text{Area of Shaded Region}=\text{Area of Hexagon}-\text{Area of Circle}$

$\text{Area of Shaded Region}=96\sqrt3-9\pi$ $\displaystyle \text{Area of Shaded Region}=96\sqrt3-9\pi$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=138.00$ $\displaystyle \text{Area of Shaded Region}=138.00$

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Example Question #117 : Hexagons

A circle is placed in a regular hexagon as shown in the figure.

If the radius of the circle is $\displaystyle 7$, then find the area of the shaded region.

Possible Answers:

425.91 $\displaystyle 425.91$

430.63 $\displaystyle 430.63$

435.28 $\displaystyle 435.28$

420.78 $\displaystyle 420.78$

Correct answer:

430.63 $\displaystyle 430.63$

Explanation:

In order to find the area of the shaded region, we must first find the areas of the hexagon and the circle.

Recall how to find the area of a regular hexagon.

$\text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(\text{side})^2$ $\displaystyle \text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(\text{side})^2$

Substitute in the length of the given side to find the area of the hexagon.

$\text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(15)^2$ $\displaystyle \text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(15)^2$

$\text{Area of Regular Hexagon}=\frac{675\sqrt3}{2}$ $\displaystyle \text{Area of Regular Hexagon}=\frac{675\sqrt3}{2}$

Next, recall how to find the area of a circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$ $\displaystyle \text{Area of Circle}=\pi\times\text{radius}^2$

Substitute in the given radius to find the area of the circle.

$\text{Area of Circle}=\pi\times (7)^2$ $\displaystyle \text{Area of Circle}=\pi\times (7)^2$

$\text{Area of Circle}=49\pi$ $\displaystyle \text{Area of Circle}=49\pi$

Now, find the area of the shaded region by subtracting the area of the circle from the area of the hexagon.

$\text{Area of Shaded Region}=\text{Area of Hexagon}-\text{Area of Circle}$ $\displaystyle \text{Area of Shaded Region}=\text{Area of Hexagon}-\text{Area of Circle}$

$\text{Area of Shaded Region}=\frac{675\sqrt3}{2}-49\pi$ $\displaystyle \text{Area of Shaded Region}=\frac{675\sqrt3}{2}-49\pi$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=430.63$ $\displaystyle \text{Area of Shaded Region}=430.63$

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Intermediate Geometry : Hexagons

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #111 : Hexagons

Example Question #12 : How To Find The Area Of A Hexagon

Example Question #112 : Hexagons

Example Question #17 : How To Find The Area Of A Hexagon

Example Question #15 : How To Find The Area Of A Hexagon

Example Question #924 : Plane Geometry

Example Question #17 : How To Find The Area Of A Hexagon

Example Question #115 : Hexagons

Example Question #116 : Hexagons

Example Question #117 : Hexagons

All Intermediate Geometry Resources

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