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Intermediate Geometry : Solid Geometry

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Example Question #1 : How To Find The Volume Of A Cube

If the side of a cube is $1\:m$ in length, what is the volume of the cube?

Possible Answers:

$4\:m^3$

$2\:m^3$

$1\:m^3$

$3\:m^3$

$6\:m^3$

Correct answer:

$1\:m^3$

Explanation:

To find the volume of a cube we just multiply a length of one of the cube's sides by itself three times, or in other words, cube it. If we call the length of a side of a cube , then:

Volume=a^3

The side of our cube is $1\:m$ in length, so we can substitute this into the equation and solve:

$Volume=a^3=(1\:m)^3=1\:m^3$

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Example Question #1 : How To Find The Volume Of A Cube

A Rubik's Cube is made up of three square layers, with nine identical smaller cubes in each layer. If one of the smaller cubes has side lenghts of 1.5 cm, what is the approximate volume of a whole Rubik's Cube?

Possible Answers:

$3.375 \; cm^3$

$91.125 \; cm^3$

$40.5 \; cm^3$

$13.5 \; cm^3$

$30.375 \; cm^3$

Correct answer:

$91.125 \; cm^3$

Explanation:

This is a great problem becaue ther are two ways to approach it. The simplest way is to realize that because each edge of the Rubik's Cube is made up of 3 of the smaller cubes we can find the edge length for the whole Rubik's Cube:

1.5 cm + 1.5 cm + 1.5 cm = 4.5 cm (edge of whole Rubik's Cube)

Now that we know the edge length finding the volume is easy, we simply multiply the length, width, and height of the cube to find the volume. This is easy because the length, width, and height are all 4.5 cm.

$4.5 \; cm \times 4.5 \; cm \times 4.5 \; cm = 91.125 \; cm^3$ (vol. of Rubik's Cube)

This is the answer. Another way to approach the problem would be by finding the volume of one of the smaller cubes, then multiplying by 9 to find the volume of one layer, then multiplying by three, because there are 3 layers.

$1.5 \; cm \times 1.5 \; cm \times 1.5 \; cm = 3.375\; cm^3$ (vol. of one small cube)

$3.375 \; cm^3 \times 9 = 30.375 \;cm^3$ (vol. of one layer of Rubik's Cube)

$30.375 \; cm^3 \times 3 = 91.125 \; cm^3$ (vol. of Rubik's Cube)

Two different methods and we got the same answer!

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Example Question #6 : How To Find The Volume Of A Cube

A cylinder is cut out of a cube as shown by the figure below.

Find the volume of the figure.

Possible Answers:

215.20

290.03

274.11

285.81

Correct answer:

285.81

Explanation:

In order to find the volume of the figure, we must first find the volumes of the cube and the cylinder.

Recall how to find the volume of a cube.

$\text{Volume of Cube}=\text{side}^3$

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

$\text{Volume of Cube}=8^3=512$

Next, recall how to find the volume of a cylinder.

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

Use the given diameter to find the length of the radius.

$\text{Radius}=\frac{\text{Diameter}}{2}=\frac{6}{2}=3$

Find the volume of the cylinder.

$\text{Volume of Cylinder}=\pi\times 3^2 \times 8=72\pi$

To find the volume of the figure, subtract the volume of the cylinder from the volume of the cube.

$\text{Volume of Figure}=512-72\pi=285.81$

Make sure to round to places after the decimal.

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Example Question #7 : How To Find The Volume Of A Cube

A cylinder is cut out of a cube as shown by the figure below.

Find the volume of the figure.

Possible Answers:

803.65

852.24

792.08

750.22

Correct answer:

803.65

Explanation:

In order to find the volume of the figure, we must first find the volumes of the cube and the cylinder.

Recall how to find the volume of a cube.

$\text{Volume of Cube}=\text{side}^3$

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

$\text{Volume of Cube}=10^3=1000$

Next, recall how to find the volume of a cylinder.

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

Use the given diameter to find the length of the radius.

$\text{Radius}=\frac{\text{Diameter}}{2}=\frac{5}{2}=2.5$

Find the volume of the cylinder.

$\text{Volume of Cylinder}=\pi\times 2.5^2 \times 10=62.5\pi$

To find the volume of the figure, subtract the volume of the cylinder from the volume of the cube.

$\text{Volume of Figure}=1000-62.5\pi=803.65$

Make sure to round to places after the decimal.

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Example Question #8 : How To Find The Volume Of A Cube

A cylinder is cut out of a cube as shown by the figure below.

Find the volume of the figure.

Possible Answers:

1447.22

1236.30

1080.69

1124.81

Correct answer:

1124.81

Explanation:

In order to find the volume of the figure, we must first find the volumes of the cube and the cylinder.

Recall how to find the volume of a cube.

$\text{Volume of Cube}=\text{side}^3$

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

$\text{Volume of Cube}=12^3=1728$

Next, recall how to find the volume of a cylinder.

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

Use the given diameter to find the length of the radius.

$\text{Radius}=\frac{\text{Diameter}}{2}=\frac{8}{2}=4$

Find the volume of the cylinder.

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

To find the volume of the figure, subtract the volume of the cylinder from the volume of the cube.

$\text{Volume of Figure}=1728-192\pi=1124.81$

Make sure to round to places after the decimal.

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Example Question #9 : How To Find The Volume Of A Cube

A cylinder is cut out of a cube as shown by the figure below.

Find the volume of the figure.

Possible Answers:

92.51

102.49

84.45

89.66

Correct answer:

89.66

Explanation:

In order to find the volume of the figure, we must first find the volumes of the cube and the cylinder.

Recall how to find the volume of a cube.

$\text{Volume of Cube}=\text{side}^3$

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

$\text{Volume of Cube}=5^3=125$

Next, recall how to find the volume of a cylinder.

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

Use the given diameter to find the length of the radius.

$\text{Radius}=\frac{\text{Diameter}}{2}=\frac{3}{2}=1.5$

Find the volume of the cylinder.

$\text{Volume of Cylinder}=\pi\times 1.5^2 \times 5=11.25\pi$

To find the volume of the figure, subtract the volume of the cylinder from the volume of the cube.

$\text{Volume of Figure}=125-11.25\pi=89.66$

Make sure to round to places after the decimal.

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Example Question #11 : How To Find The Volume Of A Cube

A cylinder is cut out of a cube as shown by the figure below.

Find the volume of the figure.

Possible Answers:

285.61

276.61

295.48

205.27

Correct answer:

276.61

Explanation:

In order to find the volume of the figure, we must first find the volumes of the cube and the cylinder.

Recall how to find the volume of a cube.

$\text{Volume of Cube}=\text{side}^3$

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

$\text{Volume of Cube}=9^3=729$

Next, recall how to find the volume of a cylinder.

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

Use the given diameter to find the length of the radius.

$\text{Radius}=\frac{\text{Diameter}}{2}=\frac{8}{2}=4$

Find the volume of the cylinder.

$\text{Volume of Cylinder}=\pi\times 4^2 \times 9=144\pi$

To find the volume of the figure, subtract the volume of the cylinder from the volume of the cube.

$\text{Volume of Figure}=729-144\pi=276.61$

Make sure to round to places after the decimal.

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Example Question #12 : How To Find The Volume Of A Cube

A cylinder is cut out of a cube as shown by the figure below.

Find the volume of the figure.

Possible Answers:

2196.90

1895.68

2258.10

2069.32

Correct answer:

2196.90

Explanation:

In order to find the volume of the figure, we must first find the volumes of the cube and the cylinder.

Recall how to find the volume of a cube.

$\text{Volume of Cube}=\text{side}^3$

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

$\text{Volume of Cube}=15^3=3375$

Next, recall how to find the volume of a cylinder.

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

Use the given diameter to find the length of the radius.

$\text{Radius}=\frac{\text{Diameter}}{2}=\frac{10}{2}=5$

Find the volume of the cylinder.

$\text{Volume of Cylinder}=\pi\times 5^2 \times 15=375\pi$

To find the volume of the figure, subtract the volume of the cylinder from the volume of the cube.

$\text{Volume of Figure}=3375-375\pi=2196.90$

Make sure to round to places after the decimal.

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Example Question #13 : How To Find The Volume Of A Cube

A cylinder is cut out of a cube as shown by the figure below.

Find the volume of the figure.

Possible Answers:

497.35

452.09

488.33

412.27

Correct answer:

497.35

Explanation:

In order to find the volume of the figure, we must first find the volumes of the cube and the cylinder.

Recall how to find the volume of a cube.

$\text{Volume of Cube}=\text{side}^3$

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

$\text{Volume of Cube}=10^3=1000$

Next, recall how to find the volume of a cylinder.

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

Use the given diameter to find the length of the radius.

$\text{Radius}=\frac{\text{Diameter}}{2}=\frac{8}{2}=4$

Find the volume of the cylinder.

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

To find the volume of the figure, subtract the volume of the cylinder from the volume of the cube.

$\text{Volume of Figure}=1000-160\pi=497.35$

Make sure to round to places after the decimal.

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Example Question #14 : How To Find The Volume Of A Cube

A cylinder is cut out of a cube as shown by the figure below.

Find the volume of the figure.

Possible Answers:

2708.58

2591.08

2596.31

2621.02

Correct answer:

2621.02

Explanation:

In order to find the volume of the figure, we must first find the volumes of the cube and the cylinder.

Recall how to find the volume of a cube.

$\text{Volume of Cube}=\text{side}^3$

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

$\text{Volume of Cube}=15^3=3375$

Next, recall how to find the volume of a cylinder.

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

Use the given diameter to find the length of the radius.

$\text{Radius}=\frac{\text{Diameter}}{2}=\frac{8}{2}=4$

Find the volume of the cylinder.

$\text{Volume of Cylinder}=\pi\times 4^2 \times 15=240\pi$

To find the volume of the figure, subtract the volume of the cylinder from the volume of the cube.

$\text{Volume of Figure}=3375-240\pi=2621.02$

Make sure to round to places after the decimal.

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Intermediate Geometry : Solid Geometry

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #1 : How To Find The Volume Of A Cube

Example Question #1 : How To Find The Volume Of A Cube

Example Question #6 : How To Find The Volume Of A Cube

Example Question #7 : How To Find The Volume Of A Cube

Example Question #8 : How To Find The Volume Of A Cube

Example Question #9 : How To Find The Volume Of A Cube

Example Question #11 : How To Find The Volume Of A Cube

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