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Intermediate Geometry : Hexagons

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Example Questions

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 , then find the area of the shaded region.

Possible Answers:

552.01

534.09

560.22

498.63

Correct answer:

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.

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

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

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

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

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

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

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

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

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

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

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

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

Solve and round to two decimal places.

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

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 , then find the area of the shaded region.

Possible Answers:

703.58

791.51

724.19

780.22

Correct answer:

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.

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

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

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

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

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

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

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

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

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

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

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

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

Solve and round to two decimal places.

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

Report an Error

Example Question #931 : Plane Geometry

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

If the radius of the circle is , then find the area of the shaded region.

Possible Answers:

800.55

791.25

784.76

806.32

Correct answer:

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.

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

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

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

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

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

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

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

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

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

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

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

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

Solve and round to two decimal places.

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

Report an Error

Example Question #27 : 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 , then find the area of the shaded region.

Possible Answers:

920.38

965.37

895.36

943.31

Correct answer:

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.

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

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

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

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

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

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

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

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

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

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

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

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

Solve and round to two decimal places.

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

Report an Error

Example Question #28 : 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 , then find the area of the shaded region.

Possible Answers:

1090.37

1240.07

1182.33

1193.67

Correct answer:

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.

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

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

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

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

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

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

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

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

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

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

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

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

Solve and round to two decimal places.

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

Report an Error

Example Question #29 : 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 , then find the area of the shaded region.

Possible Answers:

1863.39

1892.05

1885.88

1875.54

Correct answer:

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.

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

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

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

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

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

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

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

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

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

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

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

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

Solve and round to two decimal places.

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

Report an Error

Example Question #30 : 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 , find the area of the entire figure.

Possible Answers:

211.31

314.20

198.56

215.22

Correct answer:

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.

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

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

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

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

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

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

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

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

Recall how to find the area of a rectangle.

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

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

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

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

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

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

Solve and round to two decimal places.

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

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 , find the area of the entire figure.

Possible Answers:

108.92

102.49

104.95

107.51

Correct answer:

104.95

Explanation:

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

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

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

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

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

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

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

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

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

Recall how to find the area of a rectangle.

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

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

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

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

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

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

Solve and round to two decimal places.

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

Report an Error

Example Question #32 : How To Find The Area Of A Hexagon

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

If the diagonal of the hexagon is , find the area of the entire figure.

Possible Answers:

161.61

147.47

153.53

144.44

Correct answer:

153.53

Explanation:

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

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

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

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

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

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

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

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

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

Recall how to find the area of a rectangle.

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

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

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

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

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

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

Solve and round to two decimal places.

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

Report an Error

Example Question #33 : How To Find The Area Of A Hexagon

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

If the diagonal of the hexagon is , find the area of the entire figure.

Possible Answers:

839.57

812.40

875.33

901.50

Correct answer:

839.57

Explanation:

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

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

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

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

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

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

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

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

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

Recall how to find the area of a rectangle.

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

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

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

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

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

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

Solve and round to two decimal places.

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

Report an Error

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Intermediate Geometry : Hexagons

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

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

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

Example Question #931 : Plane Geometry

Example Question #27 : How To Find The Area Of A Hexagon

Example Question #28 : How To Find The Area Of A Hexagon

Example Question #29 : How To Find The Area Of A Hexagon

Example Question #30 : How To Find The Area Of A Hexagon

Example Question #932 : Plane Geometry

Example Question #32 : How To Find The Area Of A Hexagon

Example Question #33 : How To Find The Area Of A Hexagon

All Intermediate Geometry Resources

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