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Intermediate Geometry : Hexagons

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Example Questions

Example Question #21 : How To Find The Area Of A Hexagon

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

If the radius of the circle is $\displaystyle 6$, then find the area of the shaded region.

Possible Answers:

552.01 $\displaystyle 552.01$

534.09 $\displaystyle 534.09$

560.22 $\displaystyle 560.22$

498.63 $\displaystyle 498.63$

Correct answer:

552.01 $\displaystyle 552.01$

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.

$\text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(16)^2$ $\displaystyle \text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(16)^2$

$\text{Area of Regular Hexagon}=384\sqrt3$ $\displaystyle \text{Area of Regular Hexagon}=384\sqrt3$

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.

$\text{Area of Circle}=\pi\times (6)^2$ $\displaystyle \text{Area of Circle}=\pi\times (6)^2$

$\text{Area of Circle}=36\pi$ $\displaystyle \text{Area of Circle}=36\pi$

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

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

$\text{Area of Shaded Region}=384\sqrt3-36\pi$ $\displaystyle \text{Area of Shaded Region}=384\sqrt3-36\pi$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=552.01$ $\displaystyle \text{Area of Shaded Region}=552.01$

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Example Question #22 : How To Find The Area Of A Hexagon

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

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

Possible Answers:

780.22 $\displaystyle 780.22$

724.19 $\displaystyle 724.19$

703.58 $\displaystyle 703.58$

791.51 $\displaystyle 791.51$

Correct answer:

791.51 $\displaystyle 791.51$

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.

$\text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(18)^2$ $\displaystyle \text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(18)^2$

$\text{Area of Regular Hexagon}=486\sqrt3$ $\displaystyle \text{Area of Regular Hexagon}=486\sqrt3$

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.

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

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

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

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

$\text{Area of Shaded Region}=486\sqrt3-16\pi$ $\displaystyle \text{Area of Shaded Region}=486\sqrt3-16\pi$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=791.51$ $\displaystyle \text{Area of Shaded Region}=791.51$

Report an Error

Example Question #22 : How To Find The Area Of A Hexagon

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

If the radius of the circle is $\displaystyle 9$, then find the area of the shaded region.

Possible Answers:

784.76 $\displaystyle 784.76$

800.55 $\displaystyle 800.55$

791.25 $\displaystyle 791.25$

806.32 $\displaystyle 806.32$

Correct answer:

784.76 $\displaystyle 784.76$

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.

$\text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(20)^2$ $\displaystyle \text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(20)^2$

$\text{Area of Regular Hexagon}=600\sqrt3$ $\displaystyle \text{Area of Regular Hexagon}=600\sqrt3$

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.

$\text{Area of Circle}=\pi\times (9)^2$ $\displaystyle \text{Area of Circle}=\pi\times (9)^2$

$\text{Area of Circle}=81\pi$ $\displaystyle \text{Area of Circle}=81\pi$

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

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

$\text{Area of Shaded Region}=600\sqrt3-81\pi$ $\displaystyle \text{Area of Shaded Region}=600\sqrt3-81\pi$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=784.76$ $\displaystyle \text{Area of Shaded Region}=784.76$

Report an Error

Example Question #23 : How To Find The Area Of A Hexagon

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

If the radius of the circle is $\displaystyle 10$, then find the area of the shaded region.

Possible Answers:

920.38 $\displaystyle 920.38$

965.37 $\displaystyle 965.37$

943.31 $\displaystyle 943.31$

895.36 $\displaystyle 895.36$

Correct answer:

943.31 $\displaystyle 943.31$

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.

$\text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(22)^2$ $\displaystyle \text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(22)^2$

$\text{Area of Regular Hexagon}=726\sqrt3$ $\displaystyle \text{Area of Regular Hexagon}=726\sqrt3$

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.

$\text{Area of Circle}=\pi\times (10)^2$ $\displaystyle \text{Area of Circle}=\pi\times (10)^2$

$\text{Area of Circle}=100\pi$ $\displaystyle \text{Area of Circle}=100\pi$

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

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

$\text{Area of Shaded Region}=726\sqrt3-100\pi$ $\displaystyle \text{Area of Shaded Region}=726\sqrt3-100\pi$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=943.31$ $\displaystyle \text{Area of Shaded Region}=943.31$

Report an Error

Example Question #24 : How To Find The Area Of A Hexagon

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

If the radius of the circle is $\displaystyle 10$, then find the area of the shaded region.

Possible Answers:

1182.33 $\displaystyle 1182.33$

1240.07 $\displaystyle 1240.07$

1090.37 $\displaystyle 1090.37$

1193.67 $\displaystyle 1193.67$

Correct answer:

1182.33 $\displaystyle 1182.33$

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.

$\text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(24)^2$ $\displaystyle \text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(24)^2$

$\text{Area of Regular Hexagon}=864\sqrt3$ $\displaystyle \text{Area of Regular Hexagon}=864\sqrt3$

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.

$\text{Area of Circle}=\pi\times (10)^2$ $\displaystyle \text{Area of Circle}=\pi\times (10)^2$

$\text{Area of Circle}=100\pi$ $\displaystyle \text{Area of Circle}=100\pi$

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

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

$\text{Area of Shaded Region}=864\sqrt3-100\pi$ $\displaystyle \text{Area of Shaded Region}=864\sqrt3-100\pi$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=1182.33$ $\displaystyle \text{Area of Shaded Region}=1182.33$

Report an Error

Example Question #25 : How To Find The Area Of A Hexagon

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

If the radius of the circle is $\displaystyle 12$, then find the area of the shaded region.

Possible Answers:

1885.88 $\displaystyle 1885.88$

1892.05 $\displaystyle 1892.05$

1875.54 $\displaystyle 1875.54$

1863.39 $\displaystyle 1863.39$

Correct answer:

1885.88 $\displaystyle 1885.88$

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.

$\text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(30)^2$ $\displaystyle \text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(30)^2$

$\text{Area of Regular Hexagon}=1350\sqrt3$ $\displaystyle \text{Area of Regular Hexagon}=1350\sqrt3$

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.

$\text{Area of Circle}=\pi\times (12)^2$ $\displaystyle \text{Area of Circle}=\pi\times (12)^2$

$\text{Area of Circle}=144\pi$ $\displaystyle \text{Area of Circle}=144\pi$

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

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

$\text{Area of Shaded Region}=1350\sqrt3-144\pi$ $\displaystyle \text{Area of Shaded Region}=1350\sqrt3-144\pi$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=1885.88$ $\displaystyle \text{Area of Shaded Region}=1885.88$

Report an Error

Example Question #26 : How To Find The Area Of A Hexagon

A rectangle is attached to a regular hexagon as shown by the figure.

If the length of the diagonal of the hexagon is $\displaystyle 14$, find the area of the entire figure.

Possible Answers:

314.20 $\displaystyle 314.20$

211.31 $\displaystyle 211.31$

198.56 $\displaystyle 198.56$

215.22 $\displaystyle 215.22$

Correct answer:

211.31 $\displaystyle 211.31$

Explanation:

When all of the diagonals of a regular hexagon are drawn in, you should notice that the hexagon is divided into six congruent equilateral triangles. The length of the diagonal is twice the length of a side of one of the equilateral triangles.

Thus, the diagonal of a hexagon is also twice the length of a side of the regular hexagon.

$\text{Side Length of Hexagon}=\frac{\text{Diagonal of Hexagon}}{2}$ $\displaystyle \text{Side Length of Hexagon}=\frac{\text{Diagonal of Hexagon}}{2}$

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

$\text{Side Length of Hexagon}=\frac{14}{2}=7$ $\displaystyle \text{Side Length of Hexagon}=\frac{14}{2}=7$

Now, recall how to find the area of a regular hexagon.

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

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

$\text{Area of Hexagon}=\frac{3\sqrt3(7)^2}{2}=\frac{147\sqrt3}{2}$ $\displaystyle \text{Area of Hexagon}=\frac{3\sqrt3(7)^2}{2}=\frac{147\sqrt3}{2}$

Next, notice that the hexagon shares a side with the length of the rectangle.

Recall how to find the area of a rectangle.

$\text{Area of Rectangle}=\text{length}\times\text{width}$ $\displaystyle \text{Area of Rectangle}=\text{length}\times\text{width}$

Substitute in the length and the width of the rectangle to find the area.

$\text{Area of Rectangle}=7\times12=84$ $\displaystyle \text{Area of Rectangle}=7\times12=84$.

In order to find the area of the entire figure, add the areas of the hexagon and rectangle together.

$\text{Area of Entire Figure}=\text{Area of Hexagon}+\text{Area of Rectangle}$ $\displaystyle \text{Area of Entire Figure}=\text{Area of Hexagon}+\text{Area of Rectangle}$

$\text{Area of Entire Figure}=\frac{147\sqrt3}{2}+84$ $\displaystyle \text{Area of Entire Figure}=\frac{147\sqrt3}{2}+84$

Solve and round to two decimal places.

$\text{Area of Entire Figure}=211.31$ $\displaystyle \text{Area of Entire Figure}=211.31$

Report an Error

Example Question #931 : Plane Geometry

A rectangle is attached to a regular hexagon as shown by the figure.

If the diagonal of the hexagon is $\displaystyle 10$, find the area of the entire figure.

Possible Answers:

102.49 $\displaystyle 102.49$

107.51 $\displaystyle 107.51$

108.92 $\displaystyle 108.92$

104.95 $\displaystyle 104.95$

Correct answer:

104.95 $\displaystyle 104.95$

Explanation:

Thus, the diagonal of a hexagon is also twice the length of a side of the regular hexagon.

$\text{Side Length of Hexagon}=\frac{\text{Diagonal of Hexagon}}{2}$ $\displaystyle \text{Side Length of Hexagon}=\frac{\text{Diagonal of Hexagon}}{2}$

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

$\text{Side Length of Hexagon}=\frac{10}{2}=5$ $\displaystyle \text{Side Length of Hexagon}=\frac{10}{2}=5$

Now, recall how to find the area of a regular hexagon.

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

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

$\text{Area of Hexagon}=\frac{3\sqrt3(5)^2}{2}=\frac{75\sqrt3}{2}$ $\displaystyle \text{Area of Hexagon}=\frac{3\sqrt3(5)^2}{2}=\frac{75\sqrt3}{2}$

Next, notice that the hexagon shares a side with the length of the rectangle.

Recall how to find the area of a rectangle.

$\text{Area of Rectangle}=\text{length}\times\text{width}$ $\displaystyle \text{Area of Rectangle}=\text{length}\times\text{width}$

Substitute in the length and the width of the rectangle to find the area.

$\text{Area of Rectangle}=5\times 8=40$ $\displaystyle \text{Area of Rectangle}=5\times 8=40$.

In order to find the area of the entire figure, add the areas of the hexagon and rectangle together.

$\text{Area of Entire Figure}=\text{Area of Hexagon}+\text{Area of Rectangle}$ $\displaystyle \text{Area of Entire Figure}=\text{Area of Hexagon}+\text{Area of Rectangle}$

$\text{Area of Entire Figure}=\frac{75\sqrt3}{2}+40$ $\displaystyle \text{Area of Entire Figure}=\frac{75\sqrt3}{2}+40$

Solve and round to two decimal places.

$\text{Area of Entire Figure}=104.95$ $\displaystyle \text{Area of Entire Figure}=104.95$

Report an Error

Example Question #932 : Plane Geometry

A rectangle is attached to a regular hexagon as shown by the figure.

If the diagonal of the hexagon is $\displaystyle 12$, find the area of the entire figure.

Possible Answers:

147.47 $\displaystyle 147.47$

153.53 $\displaystyle 153.53$

144.44 $\displaystyle 144.44$

161.61 $\displaystyle 161.61$

Correct answer:

153.53 $\displaystyle 153.53$

Explanation:

Thus, the diagonal of a hexagon is also twice the length of a side of the regular hexagon.

$\text{Side Length of Hexagon}=\frac{\text{Diagonal of Hexagon}}{2}$ $\displaystyle \text{Side Length of Hexagon}=\frac{\text{Diagonal of Hexagon}}{2}$

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

$\text{Side Length of Hexagon}=\frac{12}{2}=6$ $\displaystyle \text{Side Length of Hexagon}=\frac{12}{2}=6$

Now, recall how to find the area of a regular hexagon.

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

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

$\text{Area of Hexagon}=\frac{3\sqrt3(6)^2}{2}=54\sqrt3$ $\displaystyle \text{Area of Hexagon}=\frac{3\sqrt3(6)^2}{2}=54\sqrt3$

Next, notice that the hexagon shares a side with the length of the rectangle.

Recall how to find the area of a rectangle.

$\text{Area of Rectangle}=\text{length}\times\text{width}$ $\displaystyle \text{Area of Rectangle}=\text{length}\times\text{width}$

Substitute in the length and the width of the rectangle to find the area.

$\text{Area of Rectangle}=6\times 10=60$ $\displaystyle \text{Area of Rectangle}=6\times 10=60$.

In order to find the area of the entire figure, add the areas of the hexagon and rectangle together.

$\text{Area of Entire Figure}=\text{Area of Hexagon}+\text{Area of Rectangle}$ $\displaystyle \text{Area of Entire Figure}=\text{Area of Hexagon}+\text{Area of Rectangle}$

$\text{Area of Entire Figure}=54\sqrt3+60$ $\displaystyle \text{Area of Entire Figure}=54\sqrt3+60$

Solve and round to two decimal places.

$\text{Area of Entire Figure}=153.53$ $\displaystyle \text{Area of Entire Figure}=153.53$

Report an Error

Example Question #933 : Plane Geometry

A rectangle is attached to a regular hexagon as shown by the figure.

If the diagonal of the hexagon is $\displaystyle 30$, find the area of the entire figure.

Possible Answers:

875.33 $\displaystyle 875.33$

839.57 $\displaystyle 839.57$

812.40 $\displaystyle 812.40$

901.50 $\displaystyle 901.50$

Correct answer:

839.57 $\displaystyle 839.57$

Explanation:

Thus, the diagonal of a hexagon is also twice the length of a side of the regular hexagon.

$\text{Side Length of Hexagon}=\frac{\text{Diagonal of Hexagon}}{2}$ $\displaystyle \text{Side Length of Hexagon}=\frac{\text{Diagonal of Hexagon}}{2}$

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

$\text{Side Length of Hexagon}=\frac{30}{2}=15$ $\displaystyle \text{Side Length of Hexagon}=\frac{30}{2}=15$

Now, recall how to find the area of a regular hexagon.

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

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

$\text{Area of Hexagon}=\frac{3\sqrt3(15)^2}{2}=\frac{675\sqrt3}{2}$ $\displaystyle \text{Area of Hexagon}=\frac{3\sqrt3(15)^2}{2}=\frac{675\sqrt3}{2}$

Next, notice that the hexagon shares a side with the length of the rectangle.

Recall how to find the area of a rectangle.

$\text{Area of Rectangle}=\text{length}\times\text{width}$ $\displaystyle \text{Area of Rectangle}=\text{length}\times\text{width}$

Substitute in the length and the width of the rectangle to find the area.

$\text{Area of Rectangle}=15\times 17=255$ $\displaystyle \text{Area of Rectangle}=15\times 17=255$.

In order to find the area of the entire figure, add the areas of the hexagon and rectangle together.

$\text{Area of Entire Figure}=\text{Area of Hexagon}+\text{Area of Rectangle}$ $\displaystyle \text{Area of Entire Figure}=\text{Area of Hexagon}+\text{Area of Rectangle}$

$\text{Area of Entire Figure}=\frac{675\sqrt3}{2}+255$ $\displaystyle \text{Area of Entire Figure}=\frac{675\sqrt3}{2}+255$

Solve and round to two decimal places.

$\text{Area of Entire Figure}=839.57$ $\displaystyle \text{Area of Entire Figure}=839.57$

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Intermediate Geometry : Hexagons

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Example Question #21 : How To Find The Area Of A Hexagon

Example Question #22 : How To Find The Area Of A Hexagon

Example Question #22 : How To Find The Area Of A Hexagon

Example Question #23 : How To Find The Area Of A Hexagon

Example Question #24 : How To Find The Area Of A Hexagon

Example Question #25 : How To Find The Area Of A Hexagon

Example Question #26 : How To Find The Area Of A Hexagon

Example Question #931 : Plane Geometry

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