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Intermediate Geometry : Solid 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 #17 : How To Find The Volume Of A Cylinder

Find the volume of a cylinder that has a radius of $\displaystyle 18$ and a height of $\frac{1}{4}$ $\displaystyle \frac{1}{4}$ .

Possible Answers:

$324\pi$ $\displaystyle 324\pi$

$81\pi$ $\displaystyle 81\pi$

$54\pi$ $\displaystyle 54\pi$

$162\pi$ $\displaystyle 162\pi$

Correct answer:

$81\pi$ $\displaystyle 81\pi$

Explanation:

Recall how to find the volume of a cylinder:

$\text{Volume of Cylinder}=\text{Area of base}\times\text{height}$ $\displaystyle \text{Volume of Cylinder}=\text{Area of base}\times\text{height}$

Since the base of a cylinder is a circle, we can write the following equation:

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

Substitute in the given values to find the volume.

$\text{Volume of Cylinder}=\pi\times 18^2\times \frac{1}{4}$ $\displaystyle \text{Volume of Cylinder}=\pi\times 18^2\times \frac{1}{4}$

Solve.

$\text{Volume of Cylinder}=81\pi$ $\displaystyle \text{Volume of Cylinder}=81\pi$

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Example Question #31 : Cylinders

Find the volume of a cylinder that has a radius of $\frac{1}{5}$ $\displaystyle \frac{1}{5}$ and a height of $\frac{2}{3}$ $\displaystyle \frac{2}{3}$ .

Possible Answers:

$\frac{1}{25}\pi$ $\displaystyle \frac{1}{25}\pi$

$\frac{1}{75}\pi$ $\displaystyle \frac{1}{75}\pi$

$\frac{4}{75}\pi$ $\displaystyle \frac{4}{75}\pi$

$\frac{2}{75}\pi$ $\displaystyle \frac{2}{75}\pi$

Correct answer:

$\frac{2}{75}\pi$ $\displaystyle \frac{2}{75}\pi$

Explanation:

Recall how to find the volume of a cylinder:

$\text{Volume of Cylinder}=\text{Area of base}\times\text{height}$ $\displaystyle \text{Volume of Cylinder}=\text{Area of base}\times\text{height}$

Since the base of a cylinder is a circle, we can write the following equation:

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

Substitute in the given values to find the volume.

$\text{Volume of Cylinder}=\pi\times (\frac{1}{5})^2\times \frac{2}{3}$ $\displaystyle \text{Volume of Cylinder}=\pi\times (\frac{1}{5})^2\times \frac{2}{3}$

Solve.

$\text{Volume of Cylinder}=\frac{2}{75}\pi$ $\displaystyle \text{Volume of Cylinder}=\frac{2}{75}\pi$

Report an Error

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

A cylinder has a smaller cylinder cut out of its core as shown by the figure below.

Find the volume of the figure.

Possible Answers:

1607.55 $\displaystyle 1607.55$

1425.32 $\displaystyle 1425.32$

1507.96 $\displaystyle 1507.96$

1236.98 $\displaystyle 1236.98$

Correct answer:

1507.96 $\displaystyle 1507.96$

Explanation:

In order to find the volume of the figure, we will first need to find the volume of both cylinders.

Recall how to find the volume of the cylinder:

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

Now, use the given radius and height to find the volume of the larger cylinder.

$\text{Volume of Larger Cylinder}=\pi\times 7^2 \times 12=588\pi$ $\displaystyle \text{Volume of Larger Cylinder}=\pi\times 7^2 \times 12=588\pi$

Next, use the given radius and height to find the volume of the smaller cylinder.

$\text{Volume of Smaller Cylinder}=\pi\times 3^2 \times 12=108\pi$ $\displaystyle \text{Volume of Smaller Cylinder}=\pi\times 3^2 \times 12=108\pi$

Subtract the volume of the smaller cylinder from the volume of the larger one to find the volume of the figure.

$\text{Volume of Figure}=588\pi-108\pi=480\pi=1507.96$ $\displaystyle \text{Volume of Figure}=588\pi-108\pi=480\pi=1507.96$

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

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

Find the volume of the figure.

Possible Answers:

3166.73 $\displaystyle 3166.73$

3020.18 $\displaystyle 3020.18$

3369.61 $\displaystyle 3369.61$

3209.58 $\displaystyle 3209.58$

Correct answer:

3166.73 $\displaystyle 3166.73$

Explanation:

In order to find the volume of the figure, we will first need to find the volume of both cylinders.

Recall how to find the volume of the cylinder:

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

Now, use the given radius and height to find the volume of the larger cylinder.

$\text{Volume of Larger Cylinder}=\pi\times 10^2 \times 12=1200\pi$ $\displaystyle \text{Volume of Larger Cylinder}=\pi\times 10^2 \times 12=1200\pi$

Next, use the given radius and height to find the volume of the smaller cylinder.

$\text{Volume of Smaller Cylinder}=\pi\times 4^2 \times 12=192\pi$ $\displaystyle \text{Volume of Smaller Cylinder}=\pi\times 4^2 \times 12=192\pi$

Subtract the volume of the smaller cylinder from the volume of the larger one to find the volume of the figure.

$\text{Volume of Figure}=1200\pi-192\pi=1008\pi=3166.73$ $\displaystyle \text{Volume of Figure}=1200\pi-192\pi=1008\pi=3166.73$

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

Report an Error

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

Find the volume of the figure.

Possible Answers:

$\displaystyle 30005.82$

$\displaystyle 29907.96$

$\displaystyle 28800.64$

$\displaystyle 29632.23$

Correct answer:

$\displaystyle 29907.96$

Explanation:

In order to find the volume of the figure, we will first need to find the volume of both cylinders.

Recall how to find the volume of the cylinder:

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

Now, use the given radius and height to find the volume of the larger cylinder.

$\text{Volume of Larger Cylinder}=\pi\times 21^2 \times 35=15435\pi$ $\displaystyle \text{Volume of Larger Cylinder}=\pi\times 21^2 \times 35=15435\pi$

Next, use the given radius and height to find the volume of the smaller cylinder.

$\text{Volume of Smaller Cylinder}=\pi\times 13^2 \times 35=5915\pi$ $\displaystyle \text{Volume of Smaller Cylinder}=\pi\times 13^2 \times 35=5915\pi$

Subtract the volume of the smaller cylinder from the volume of the larger one to find the volume of the figure.

$\text{Volume of Figure}=15435\pi-5915\pi=9520\pi=29907.96$ $\displaystyle \text{Volume of Figure}=15435\pi-5915\pi=9520\pi=29907.96$

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

Report an Error

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

Find the volume of the figure.

Possible Answers:

$\displaystyle 26551.22$

$\displaystyle 21635.08$

$\displaystyle 27159.63$

$\displaystyle 26389.38$

Correct answer:

$\displaystyle 26389.38$

Explanation:

In order to find the volume of the figure, we will first need to find the volume of both cylinders.

Recall how to find the volume of the cylinder:

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

Now, use the given radius and height to find the volume of the larger cylinder.

$\text{Volume of Larger Cylinder}=\pi\times 20^2 \times 25=10000\pi$ $\displaystyle \text{Volume of Larger Cylinder}=\pi\times 20^2 \times 25=10000\pi$

Next, use the given radius and height to find the volume of the smaller cylinder.

$\text{Volume of Smaller Cylinder}=\pi\times 8^2 \times 25=1600\pi$ $\displaystyle \text{Volume of Smaller Cylinder}=\pi\times 8^2 \times 25=1600\pi$

Subtract the volume of the smaller cylinder from the volume of the larger one to find the volume of the figure.

$\text{Volume of Figure}=10000\pi-1600\pi=8400\pi=26389.38$ $\displaystyle \text{Volume of Figure}=10000\pi-1600\pi=8400\pi=26389.38$

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

Report an Error

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

Find the volume of the figure.

Possible Answers:

$\displaystyle 15966.32$

$\displaystyle 14844.03$

$\displaystyle 15633.08$

$\displaystyle 13222.07$

Correct answer:

$\displaystyle 14844.03$

Explanation:

In order to find the volume of the figure, we will first need to find the volume of both cylinders.

Recall how to find the volume of the cylinder:

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

Now, use the given radius and height to find the volume of the larger cylinder.

$\text{Volume of Larger Cylinder}=\pi\times 15^2 \times 25=5625\pi$ $\displaystyle \text{Volume of Larger Cylinder}=\pi\times 15^2 \times 25=5625\pi$

Next, use the given radius and height to find the volume of the smaller cylinder.

$\text{Volume of Smaller Cylinder}=\pi\times 6^2 \times 25=900\pi$ $\displaystyle \text{Volume of Smaller Cylinder}=\pi\times 6^2 \times 25=900\pi$

Subtract the volume of the smaller cylinder from the volume of the larger one to find the volume of the figure.

$\text{Volume of Figure}=5625\pi-900\pi=4725\pi=14844.03$ $\displaystyle \text{Volume of Figure}=5625\pi-900\pi=4725\pi=14844.03$

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

Report an Error

Example Question #111 : Solid Geometry

The figure below represents a cylinder with a smaller cylinder removed from its middle.

Find the volume of the figure.

Possible Answers:

1896.77 $\displaystyle 1896.77$

1850.36 $\displaystyle 1850.36$

1865.32 $\displaystyle 1865.32$

1884.96 $\displaystyle 1884.96$

Correct answer:

1884.96 $\displaystyle 1884.96$

Explanation:

In order to find the volume of the figure, we will first need to find the volume of both cylinders.

Recall how to find the volume of the cylinder:

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

Now, use the given radius and height to find the volume of the larger cylinder.

$\text{Volume of Larger Cylinder}=\pi\times 8^2 \times 10=640\pi$ $\displaystyle \text{Volume of Larger Cylinder}=\pi\times 8^2 \times 10=640\pi$

Next, use the given radius and height to find the volume of the smaller cylinder.

$\text{Volume of Smaller Cylinder}=\pi\times 2^2 \times 10=40\pi$ $\displaystyle \text{Volume of Smaller Cylinder}=\pi\times 2^2 \times 10=40\pi$

Subtract the volume of the smaller cylinder from the volume of the larger one to find the volume of the figure.

$\text{Volume of Figure}=640\pi-40\pi=600\pi=1884.96$ $\displaystyle \text{Volume of Figure}=640\pi-40\pi=600\pi=1884.96$

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

Report an Error

Example Question #851 : Sat Mathematics

Find the volume of the figure.

Possible Answers:

2036.33 $\displaystyle 2036.33$

2073.45 $\displaystyle 2073.45$

2158.25 $\displaystyle 2158.25$

2200.94 $\displaystyle 2200.94$

Correct answer:

2073.45 $\displaystyle 2073.45$

Explanation:

In order to find the volume of the figure, we will first need to find the volume of both cylinders.

Recall how to find the volume of the cylinder:

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

Now, use the given radius and height to find the volume of the larger cylinder.

$\text{Volume of Larger Cylinder}=\pi\times 8^2 \times 12=768\pi$ $\displaystyle \text{Volume of Larger Cylinder}=\pi\times 8^2 \times 12=768\pi$

Next, use the given radius and height to find the volume of the smaller cylinder.

$\text{Volume of Smaller Cylinder}=\pi\times 3^2 \times 12=108\pi$ $\displaystyle \text{Volume of Smaller Cylinder}=\pi\times 3^2 \times 12=108\pi$

Subtract the volume of the smaller cylinder from the volume of the larger one to find the volume of the figure.

$\text{Volume of Figure}=768\pi-108\pi=660\pi=2073.45$ $\displaystyle \text{Volume of Figure}=768\pi-108\pi=660\pi=2073.45$

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

Report an Error

Example Question #113 : Solid Geometry

Find the volume of the figure.

Possible Answers:

1006.37 $\displaystyle 1006.37$

1225.22 $\displaystyle 1225.22$

1109.87 $\displaystyle 1109.87$

1206.37 $\displaystyle 1206.37$

Correct answer:

1206.37 $\displaystyle 1206.37$

Explanation:

In order to find the volume of the figure, we will first need to find the volume of both cylinders.

Recall how to find the volume of the cylinder:

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

Now, use the given radius and height to find the volume of the larger cylinder.

$\text{Volume of Larger Cylinder}=\pi\times 6^2 \times 12=432\pi$ $\displaystyle \text{Volume of Larger Cylinder}=\pi\times 6^2 \times 12=432\pi$

Next, use the given radius and height to find the volume of the smaller cylinder.

$\text{Volume of Smaller Cylinder}=\pi\times 2^2 \times 12=48\pi$ $\displaystyle \text{Volume of Smaller Cylinder}=\pi\times 2^2 \times 12=48\pi$

Subtract the volume of the smaller cylinder from the volume of the larger one to find the volume of the figure.

$\text{Volume of Figure}=432\pi-48\pi=384\pi=1206.37$ $\displaystyle \text{Volume of Figure}=432\pi-48\pi=384\pi=1206.37$

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

Report an Error

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Intermediate Geometry : Solid Geometry

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #17 : How To Find The Volume Of A Cylinder

Example Question #31 : Cylinders

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

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

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

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

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

Example Question #111 : Solid Geometry

Example Question #851 : Sat Mathematics

Example Question #113 : Solid Geometry

All Intermediate Geometry Resources

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