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Intermediate Geometry : Intermediate Geometry

Study concepts, example questions & explanations for Intermediate Geometry

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8 Diagnostic Tests 250 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #21 : Spheres

A sphere is cut in half and placed on top of a cylinder so that they share the same base as shown by the figure below.

Find the volume of the figure.

Possible Answers:

618.32 $\displaystyle 618.32$

650.12 $\displaystyle 650.12$

679.05 $\displaystyle 679.05$

636.70 $\displaystyle 636.70$

Correct answer:

636.70 $\displaystyle 636.70$

Explanation:

In order to find the volume of the figure, we will first need to find the volumes of the cylinder and of the half sphere.

Recall how to find the volume of a cylinder:

$\text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$ $\displaystyle \text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$

$\text{Volume of Cylinder}=\pi\times 4^2 \times 10=160\pi$ $\displaystyle \text{Volume of Cylinder}=\pi\times 4^2 \times 10=160\pi$

Next, find the volume of the half sphere.

$\text{Volume of Half sphere}=\frac{1}{2}(\frac{4}{3}\pi\times\text{radius}^3)=\frac{2}{3}\pi\times\text{radius}^3$ $\displaystyle \text{Volume of Half sphere}=\frac{1}{2}(\frac{4}{3}\pi\times\text{radius}^3)=\frac{2}{3}\pi\times\text{radius}^3$

Plug in the radius to find the volume of the half sphere.

$\text{Volume of Half Sphere}=\frac{2}{3}\times\pi\times 4^3=\frac{128}{3}\pi$ $\displaystyle \text{Volume of Half Sphere}=\frac{2}{3}\times\pi\times 4^3=\frac{128}{3}\pi$

Next, add up the two volumes together to find the volume of the figure.

$\text{Volume of Figure}=\text{Volume of Cylinder}+\text{Volume of Half Sphere}$ $\displaystyle \text{Volume of Figure}=\text{Volume of Cylinder}+\text{Volume of Half Sphere}$

$\text{Volume of Figure}=160\pi + \frac{128}{3}\pi=636.70$ $\displaystyle \text{Volume of Figure}=160\pi + \frac{128}{3}\pi=636.70$

Remember to round to $\displaystyle 2$ places after the decimal.

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Example Question #251 : Solid Geometry

A sphere is cut in half and placed on top of a cylinder so that they share the same base as shown by the figure below.

Find the volume of the figure.

Possible Answers:

5333.56 $\displaystyle 5333.56$

5190.20 $\displaystyle 5190.20$

5294.63 $\displaystyle 5294.63$

4921.59 $\displaystyle 4921.59$

Correct answer:

5294.63 $\displaystyle 5294.63$

Explanation:

In order to find the volume of the figure, we will first need to find the volumes of the cylinder and of the half sphere.

Recall how to find the volume of a cylinder:

$\text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$ $\displaystyle \text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$

$\text{Volume of Cylinder}=\pi\times 8^2 \times 21=1344\pi$ $\displaystyle \text{Volume of Cylinder}=\pi\times 8^2 \times 21=1344\pi$

Next, find the volume of the half sphere.

Plug in the radius to find the volume of the half sphere.

$\text{Volume of Half Sphere}=\frac{2}{3}\times\pi\times 8^3=\frac{1024}{3}\pi$ $\displaystyle \text{Volume of Half Sphere}=\frac{2}{3}\times\pi\times 8^3=\frac{1024}{3}\pi$

Next, add up the two volumes together to find the volume of the figure.

$\text{Volume of Figure}=\text{Volume of Cylinder}+\text{Volume of Half Sphere}$ $\displaystyle \text{Volume of Figure}=\text{Volume of Cylinder}+\text{Volume of Half Sphere}$

$\text{Volume of Figure}=1344\pi + \frac{1024}{3}\pi=5294.63$ $\displaystyle \text{Volume of Figure}=1344\pi + \frac{1024}{3}\pi=5294.63$

Remember to round to $\displaystyle 2$ places after the decimal.

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Example Question #23 : How To Find The Volume Of A Sphere

A sphere is cut in half and placed on top of a cylinder so that they share the same base as shown by the figure below.

Find the volume of the figure.

Possible Answers:

109.92 $\displaystyle 109.92$

124.12 $\displaystyle 124.12$

131.09 $\displaystyle 131.09$

129.85 $\displaystyle 129.85$

Correct answer:

129.85 $\displaystyle 129.85$

Explanation:

In order to find the volume of the figure, we will first need to find the volumes of the cylinder and of the half sphere.

Recall how to find the volume of a cylinder:

$\text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$ $\displaystyle \text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$

$\text{Volume of Cylinder}=\pi\times 2^2 \times 9=36\pi$ $\displaystyle \text{Volume of Cylinder}=\pi\times 2^2 \times 9=36\pi$

Next, find the volume of the half sphere.

Plug in the radius to find the volume of the half sphere.

$\text{Volume of Half Sphere}=\frac{2}{3}\times\pi\times 2^3=\frac{16}{3}\pi$ $\displaystyle \text{Volume of Half Sphere}=\frac{2}{3}\times\pi\times 2^3=\frac{16}{3}\pi$

Next, add up the two volumes together to find the volume of the figure.

$\text{Volume of Figure}=\text{Volume of Cylinder}+\text{Volume of Half Sphere}$ $\displaystyle \text{Volume of Figure}=\text{Volume of Cylinder}+\text{Volume of Half Sphere}$

$\text{Volume of Figure}=36\pi + \frac{16}{3}\pi=129.85$ $\displaystyle \text{Volume of Figure}=36\pi + \frac{16}{3}\pi=129.85$

Remember to round to $\displaystyle 2$ places after the decimal.

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Example Question #1221 : Intermediate Geometry

A sphere is cut in half and placed on top of a cylinder so that they share the same base as shown by the figure below.

Find the volume of the figure.

Possible Answers:

258.10 $\displaystyle 258.10$

249.88 $\displaystyle 249.88$

254.47 $\displaystyle 254.47$

249.31 $\displaystyle 249.31$

Correct answer:

254.47 $\displaystyle 254.47$

Explanation:

In order to find the volume of the figure, we will first need to find the volumes of the cylinder and of the half sphere.

Recall how to find the volume of a cylinder:

$\text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$ $\displaystyle \text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$

$\text{Volume of Cylinder}=\pi\times 3^2 \times 7=63\pi$ $\displaystyle \text{Volume of Cylinder}=\pi\times 3^2 \times 7=63\pi$

Next, find the volume of the half sphere.

Plug in the radius to find the volume of the half sphere.

$\text{Volume of Half Sphere}=\frac{2}{3}\times\pi\times 3^3=18\pi$ $\displaystyle \text{Volume of Half Sphere}=\frac{2}{3}\times\pi\times 3^3=18\pi$

Next, add up the two volumes together to find the volume of the figure.

$\text{Volume of Figure}=\text{Volume of Cylinder}+\text{Volume of Half Sphere}$ $\displaystyle \text{Volume of Figure}=\text{Volume of Cylinder}+\text{Volume of Half Sphere}$

$\text{Volume of Figure}=63\pi + 18\pi=254.47$ $\displaystyle \text{Volume of Figure}=63\pi + 18\pi=254.47$

Remember to round to $\displaystyle 2$ places after the decimal.

Report an Error

Example Question #21 : How To Find The Volume Of A Sphere

A sphere is cut in half and placed on top of a cylinder so that they share the same base as shown by the figure below.

Find the volume of the figure.

Possible Answers:

242.95 $\displaystyle 242.95$

281.52 $\displaystyle 281.52$

234.73 $\displaystyle 234.73$

214.20 $\displaystyle 214.20$

Correct answer:

242.95 $\displaystyle 242.95$

Explanation:

In order to find the volume of the figure, we will first need to find the volumes of the cylinder and of the half sphere.

Recall how to find the volume of a cylinder:

$\text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$ $\displaystyle \text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$

$\text{Volume of Cylinder}=\pi\times 2^2 \times 18=72\pi$ $\displaystyle \text{Volume of Cylinder}=\pi\times 2^2 \times 18=72\pi$

Next, find the volume of the half sphere.

Plug in the radius to find the volume of the half sphere.

$\text{Volume of Half Sphere}=\frac{2}{3}\times\pi\times 2^3=\frac{16}{3}\pi$ $\displaystyle \text{Volume of Half Sphere}=\frac{2}{3}\times\pi\times 2^3=\frac{16}{3}\pi$

Next, add up the two volumes together to find the volume of the figure.

$\text{Volume of Figure}=\text{Volume of Cylinder}+\text{Volume of Half Sphere}$ $\displaystyle \text{Volume of Figure}=\text{Volume of Cylinder}+\text{Volume of Half Sphere}$

$\text{Volume of Figure}=72\pi + \frac{16}{3}\pi=242.95$ $\displaystyle \text{Volume of Figure}=72\pi + \frac{16}{3}\pi=242.95$

Remember to round to $\displaystyle 2$ places after the decimal.

Report an Error

Example Question #21 : How To Find The Volume Of A Sphere

A sphere is cut in half and placed on top of a cylinder so that they share the same base as shown by the figure below.

Find the volume of the figure.

Possible Answers:

17.80 $\displaystyle 17.80$

15.79 $\displaystyle 15.79$

13.44 $\displaystyle 13.44$

12.49 $\displaystyle 12.49$

Correct answer:

17.80 $\displaystyle 17.80$

Explanation:

In order to find the volume of the figure, we will first need to find the volumes of the cylinder and of the half sphere.

Recall how to find the volume of a cylinder:

$\text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$ $\displaystyle \text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$

$\text{Volume of Cylinder}=\pi\times 1^2 \times 5=5\pi$ $\displaystyle \text{Volume of Cylinder}=\pi\times 1^2 \times 5=5\pi$

Next, find the volume of the half sphere.

Plug in the radius to find the volume of the half sphere.

$\text{Volume of Half Sphere}=\frac{2}{3}\times\pi\times 1^3=\frac{2}{3}\pi$ $\displaystyle \text{Volume of Half Sphere}=\frac{2}{3}\times\pi\times 1^3=\frac{2}{3}\pi$

Next, add up the two volumes together to find the volume of the figure.

$\text{Volume of Figure}=\text{Volume of Cylinder}+\text{Volume of Half Sphere}$ $\displaystyle \text{Volume of Figure}=\text{Volume of Cylinder}+\text{Volume of Half Sphere}$

$\text{Volume of Figure}=5\pi + \frac{2}{3}\pi=17.80$ $\displaystyle \text{Volume of Figure}=5\pi + \frac{2}{3}\pi=17.80$

Remember to round to $\displaystyle 2$ places after the decimal.

Report an Error

Example Question #21 : Spheres

A sphere is cut in half and placed on top of a cylinder so that they share the same base as shown by the figure below.

Find the volume of the figure.

Possible Answers:

8.39 $\displaystyle 8.39$

12.59 $\displaystyle 12.59$

10.44 $\displaystyle 10.44$

11.52 $\displaystyle 11.52$

Correct answer:

11.52 $\displaystyle 11.52$

Explanation:

In order to find the volume of the figure, we will first need to find the volumes of the cylinder and of the half sphere.

Recall how to find the volume of a cylinder:

$\text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$ $\displaystyle \text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$

$\text{Volume of Cylinder}=\pi\times 1^2 \times 3=3\pi$ $\displaystyle \text{Volume of Cylinder}=\pi\times 1^2 \times 3=3\pi$

Next, find the volume of the half sphere.

Plug in the radius to find the volume of the half sphere.

$\text{Volume of Half Sphere}=\frac{2}{3}\times\pi\times 1^3=\frac{2}{3}\pi$ $\displaystyle \text{Volume of Half Sphere}=\frac{2}{3}\times\pi\times 1^3=\frac{2}{3}\pi$

Next, add up the two volumes together to find the volume of the figure.

$\text{Volume of Figure}=\text{Volume of Cylinder}+\text{Volume of Half Sphere}$ $\displaystyle \text{Volume of Figure}=\text{Volume of Cylinder}+\text{Volume of Half Sphere}$

$\text{Volume of Figure}=3\pi + \frac{2}{3}\pi=11.52$ $\displaystyle \text{Volume of Figure}=3\pi + \frac{2}{3}\pi=11.52$

Remember to round to $\displaystyle 2$ places after the decimal.

Report an Error

Example Question #27 : How To Find The Volume Of A Sphere

A sphere is cut in half and placed on top of a cylinder so that they share the same base as shown by the figure below.

Find the volume of the figure.

Possible Answers:

938.29 $\displaystyle 938.29$

945.58 $\displaystyle 945.58$

922.45 $\displaystyle 922.45$

990.05 $\displaystyle 990.05$

Correct answer:

938.29 $\displaystyle 938.29$

Explanation:

In order to find the volume of the figure, we will first need to find the volumes of the cylinder and of the half sphere.

Recall how to find the volume of a cylinder:

$\text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$ $\displaystyle \text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$

$\text{Volume of Cylinder}=\pi\times 4^2 \times 16=256\pi$ $\displaystyle \text{Volume of Cylinder}=\pi\times 4^2 \times 16=256\pi$

Next, find the volume of the half sphere.

Plug in the radius to find the volume of the half sphere.

$\text{Volume of Half Sphere}=\frac{2}{3}\times\pi\times 4^3=\frac{128}{3}\pi$ $\displaystyle \text{Volume of Half Sphere}=\frac{2}{3}\times\pi\times 4^3=\frac{128}{3}\pi$

Next, add up the two volumes together to find the volume of the figure.

$\text{Volume of Figure}=\text{Volume of Cylinder}+\text{Volume of Half Sphere}$ $\displaystyle \text{Volume of Figure}=\text{Volume of Cylinder}+\text{Volume of Half Sphere}$

$\text{Volume of Figure}=256\pi + \frac{128}{3}\pi=938.29$ $\displaystyle \text{Volume of Figure}=256\pi + \frac{128}{3}\pi=938.29$

Remember to round to $\displaystyle 2$ places after the decimal.

Report an Error

Example Question #28 : How To Find The Volume Of A Sphere

A sphere is cut in half and placed on top of a cylinder so that they share the same base as shown by the figure below.

Find the volume of the figure.

Possible Answers:

8224.57 $\displaystyle 8224.57$

8667.79 $\displaystyle 8667.79$

8377.58 $\displaystyle 8377.58$

8484.12 $\displaystyle 8484.12$

Correct answer:

8377.58 $\displaystyle 8377.58$

Explanation:

In order to find the volume of the figure, we will first need to find the volumes of the cylinder and of the half sphere.

Recall how to find the volume of a cylinder:

$\text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$ $\displaystyle \text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$

$\text{Volume of Cylinder}=\pi\times 10^2 \times 20=2000\pi$ $\displaystyle \text{Volume of Cylinder}=\pi\times 10^2 \times 20=2000\pi$

Next, find the volume of the half sphere.

Plug in the radius to find the volume of the half sphere.

$\text{Volume of Half Sphere}=\frac{2}{3}\times\pi\times 10^3=\frac{2000}{3}\pi$ $\displaystyle \text{Volume of Half Sphere}=\frac{2}{3}\times\pi\times 10^3=\frac{2000}{3}\pi$

Next, add up the two volumes together to find the volume of the figure.

$\text{Volume of Figure}=\text{Volume of Cylinder}+\text{Volume of Half Sphere}$ $\displaystyle \text{Volume of Figure}=\text{Volume of Cylinder}+\text{Volume of Half Sphere}$

$\text{Volume of Figure}=2000\pi + \frac{2000}{3}\pi=8377.58$ $\displaystyle \text{Volume of Figure}=2000\pi + \frac{2000}{3}\pi=8377.58$

Remember to round to $\displaystyle 2$ places after the decimal.

Report an Error

Example Question #21 : Spheres

A sphere with a radius of $\frac{3}{4}$ $\displaystyle \frac{3}{4}$ is cut out of a cube that has a side length of $\displaystyle 2$. What is the volume of the resulting figure?

Possible Answers:

7.61 $\displaystyle 7.61$

6.09 $\displaystyle 6.09$

5.51 $\displaystyle 5.51$

6.23 $\displaystyle 6.23$

Correct answer:

6.23 $\displaystyle 6.23$

Explanation:

Since the cube is bigger, we will be subtracting the volume of the sphere from the volume of the cube.

Start by recalling how to find the volume of a sphere.

$\text{Volume of Sphere}=\frac{4}{3}\pi r^3$ $\displaystyle \text{Volume of Sphere}=\frac{4}{3}\pi r^3$

Plug in the given radius to find the volume.

$\text{Volume of Sphere}=\frac{4}{3}\pi(\frac{3}{4})^3=\frac{9}{16}\pi$ $\displaystyle \text{Volume of Sphere}=\frac{4}{3}\pi(\frac{3}{4})^3=\frac{9}{16}\pi$

Next, recall how to find the volume of a cube:

$\text{Volume of Cube}=(\text{edge})^3$ $\displaystyle \text{Volume of Cube}=(\text{edge})^3$

Plug in the given side length to find the volume of the cube.

$\text{Volume of Cube}=2^3=8$ $\displaystyle \text{Volume of Cube}=2^3=8$

Finally, subtract the volume of the sphere from the volume of the cube.

$\text{Volume of Figure}=8-\frac{9}{16}\pi=6.23$ $\displaystyle \text{Volume of Figure}=8-\frac{9}{16}\pi=6.23$

Make sure to round to $\displaystyle 2$ places after the decimal.

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Intermediate Geometry : Intermediate Geometry

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #21 : Spheres

Example Question #251 : Solid Geometry

Example Question #23 : How To Find The Volume Of A Sphere

Example Question #1221 : Intermediate Geometry

Example Question #21 : How To Find The Volume Of A Sphere

Example Question #21 : How To Find The Volume Of A Sphere

Example Question #21 : Spheres

Example Question #27 : How To Find The Volume Of A Sphere

Example Question #28 : How To Find The Volume Of A Sphere

Example Question #21 : Spheres

All Intermediate Geometry Resources

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