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Intermediate Geometry : Cubes

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Example Questions

Example Question #25 : How To Find The Volume Of A Cube

A rectangular prism with a square base is cut out of a cube as shown by the figure below.

Find the volume of the figure.

Possible Answers:

2615 $\displaystyle 2615$

2765 $\displaystyle 2765$

2835 $\displaystyle 2835$

2995 $\displaystyle 2995$

Correct answer:

2835 $\displaystyle 2835$

Explanation:

Start by finding the volume of the cube.

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

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

Next, find the volume of the rectangular prism.

$\text{Volume of Rectangular Prism}=\text{width}\times\text{length}\times\text{height}$ $\displaystyle \text{Volume of Rectangular Prism}=\text{width}\times\text{length}\times\text{height}$

$\text{Volume of Rectangular Prism}=6 \times 6 \times 15=540$ $\displaystyle \text{Volume of Rectangular Prism}=6 \times 6 \times 15=540$

Subtract the volume of the rectangular prism from that of the cube to find the volume of the figure.

$\text{Volume of Figure}=\text{Volume of Cube}-\text{Volume of Rectangular Prism}$ $\displaystyle \text{Volume of Figure}=\text{Volume of Cube}-\text{Volume of Rectangular Prism}$

$\text{Volume of Figure}=3375- 540=2835$ $\displaystyle \text{Volume of Figure}=3375- 540=2835$

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Example Question #26 : How To Find The Volume Of A Cube

A rectangular prism with a square base is cut out of a cube as shown by the figure below.

Find the volume of the figure.

Possible Answers:

1140 $\displaystyle 1140$

1200 $\displaystyle 1200$

880 $\displaystyle 880$

990 $\displaystyle 990$

Correct answer:

1140 $\displaystyle 1140$

Explanation:

Start by finding the volume of the cube.

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

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

Next, find the volume of the rectangular prism.

$\text{Volume of Rectangular Prism}=\text{width}\times\text{length}\times\text{height}$ $\displaystyle \text{Volume of Rectangular Prism}=\text{width}\times\text{length}\times\text{height}$

$\text{Volume of Rectangular Prism}=7 \times 7 \times 12=588$ $\displaystyle \text{Volume of Rectangular Prism}=7 \times 7 \times 12=588$

Subtract the volume of the rectangular prism from that of the cube to find the volume of the figure.

$\text{Volume of Figure}=\text{Volume of Cube}-\text{Volume of Rectangular Prism}$ $\displaystyle \text{Volume of Figure}=\text{Volume of Cube}-\text{Volume of Rectangular Prism}$

$\text{Volume of Figure}=1728 - 588=1140$ $\displaystyle \text{Volume of Figure}=1728 - 588=1140$

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

A rectangular prism with a square base is cut out of a cube as shown by the figure below.

Find the volume of the figure.

Possible Answers:

960 $\displaystyle 960$

1420 $\displaystyle 1420$

1020 $\displaystyle 1020$

810 $\displaystyle 810$

Correct answer:

960 $\displaystyle 960$

Explanation:

Start by finding the volume of the cube.

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

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

Next, find the volume of the rectangular prism.

$\text{Volume of Rectangular Prism}=\text{width}\times\text{length}\times\text{height}$ $\displaystyle \text{Volume of Rectangular Prism}=\text{width}\times\text{length}\times\text{height}$

$\text{Volume of Rectangular Prism}=8 \times 8 \times 12=768$ $\displaystyle \text{Volume of Rectangular Prism}=8 \times 8 \times 12=768$

Subtract the volume of the rectangular prism from that of the cube to find the volume of the figure.

$\text{Volume of Figure}=\text{Volume of Cube}-\text{Volume of Rectangular Prism}$ $\displaystyle \text{Volume of Figure}=\text{Volume of Cube}-\text{Volume of Rectangular Prism}$

$\text{Volume of Figure}=1728 - 768 =960$ $\displaystyle \text{Volume of Figure}=1728 - 768 =960$

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

A rectangular prism with a square base is cut out of a cube as shown by the figure below.

Find the volume of the figure.

Possible Answers:

2304 $\displaystyle 2304$

2222 $\displaystyle 2222$

2408 $\displaystyle 2408$

2296 $\displaystyle 2296$

Correct answer:

2304 $\displaystyle 2304$

Explanation:

Start by finding the volume of the cube.

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

$\text{Volume of Cube}=18^3=5832$ $\displaystyle \text{Volume of Cube}=18^3=5832$

Next, find the volume of the rectangular prism.

$\text{Volume of Rectangular Prism}=\text{width}\times\text{length}\times\text{height}$ $\displaystyle \text{Volume of Rectangular Prism}=\text{width}\times\text{length}\times\text{height}$

$\text{Volume of Rectangular Prism}=14 \times 14 \times 18=3528$ $\displaystyle \text{Volume of Rectangular Prism}=14 \times 14 \times 18=3528$

Subtract the volume of the rectangular prism from that of the cube to find the volume of the figure.

$\text{Volume of Figure}=\text{Volume of Cube}-\text{Volume of Rectangular Prism}$ $\displaystyle \text{Volume of Figure}=\text{Volume of Cube}-\text{Volume of Rectangular Prism}$

$\text{Volume of Figure}=5832- 3528=2304$ $\displaystyle \text{Volume of Figure}=5832- 3528=2304$

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Example Question #29 : How To Find The Volume Of A Cube

A rectangular prism with a square base is cut out of the cube as shown by the figure below.

Find the volume of the figure.

Possible Answers:

20250 $\displaystyle 20250$

19005 $\displaystyle 19005$

18920 $\displaystyle 18920$

17750 $\displaystyle 17750$

Correct answer:

20250 $\displaystyle 20250$

Explanation:

Start by finding the volume of the cube.

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

$\text{Volume of Cube}=30^3=27000$ $\displaystyle \text{Volume of Cube}=30^3=27000$

Next, find the volume of the rectangular prism.

$\text{Volume of Rectangular Prism}=\text{width}\times\text{length}\times\text{height}$ $\displaystyle \text{Volume of Rectangular Prism}=\text{width}\times\text{length}\times\text{height}$

$\text{Volume of Rectangular Prism}=15 \times15 \times 30=6750$ $\displaystyle \text{Volume of Rectangular Prism}=15 \times15 \times 30=6750$

Subtract the volume of the rectangular prism from that of the cube to find the volume of the figure.

$\text{Volume of Figure}=\text{Volume of Cube}-\text{Volume of Rectangular Prism}$ $\displaystyle \text{Volume of Figure}=\text{Volume of Cube}-\text{Volume of Rectangular Prism}$

$\text{Volume of Figure}=27000- 6750=20250$ $\displaystyle \text{Volume of Figure}=27000- 6750=20250$

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Example Question #30 : How To Find The Volume Of A Cube

A sphere with a radius of $\displaystyle 2$ is cut out of a cube that has a side edge of $\displaystyle 4$. What is the volume of the resulting shape?

Possible Answers:

25.10 $\displaystyle 25.10$

34.11 $\displaystyle 34.11$

26.66 $\displaystyle 26.66$

30.49 $\displaystyle 30.49$

Correct answer:

30.49 $\displaystyle 30.49$

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.

$\text{Volume of Sphere}=\frac{4}{3}\pi(2)^3=\frac{32}{3}\pi$ $\displaystyle \text{Volume of Sphere}=\frac{4}{3}\pi(2)^3=\frac{32}{3}\pi$

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.

$\text{Volume of Cube}=4^3=64$ $\displaystyle \text{Volume of Cube}=4^3=64$

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

$\text{Volume of Figure}=64-\frac{32}{3}\pi=30.49$ $\displaystyle \text{Volume of Figure}=64-\frac{32}{3}\pi=30.49$

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

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Example Question #31 : How To Find The Volume Of A Cube

A sphere with a radius of $\displaystyle 1$ is cut out of a cube that has a side length of $\displaystyle 5$. What is the volume of the resulting figure?

Possible Answers:

122.12 $\displaystyle 122.12$

120.81 $\displaystyle 120.81$

109.58 $\displaystyle 109.58$

106.65 $\displaystyle 106.65$

Correct answer:

120.81 $\displaystyle 120.81$

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.

$\text{Volume of Sphere}=\frac{4}{3}\pi(1)^3=\frac{4}{3}\pi$ $\displaystyle \text{Volume of Sphere}=\frac{4}{3}\pi(1)^3=\frac{4}{3}\pi$

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.

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

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

$\text{Volume of Figure}=125-\frac{4}{3}\pi=120.81$ $\displaystyle \text{Volume of Figure}=125-\frac{4}{3}\pi=120.81$

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

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Example Question #32 : How To Find The Volume Of A Cube

A sphere with a radius of $\displaystyle 3$ is cut out of a cube with a side length of $\displaystyle 6$. What is the volume of the resulting figure?

Possible Answers:

102.90 $\displaystyle 102.90$

98.20 $\displaystyle 98.20$

101.21 $\displaystyle 101.21$

110.51 $\displaystyle 110.51$

Correct answer:

102.90 $\displaystyle 102.90$

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.

$\text{Volume of Sphere}=\frac{4}{3}\pi(3)^3=36\pi$ $\displaystyle \text{Volume of Sphere}=\frac{4}{3}\pi(3)^3=36\pi$

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.

$\text{Volume of Cube}=6^3=216$ $\displaystyle \text{Volume of Cube}=6^3=216$

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

$\text{Volume of Figure}=216-36\pi=102.90$ $\displaystyle \text{Volume of Figure}=216-36\pi=102.90$

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

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Example Question #33 : How To Find The Volume Of A Cube

A sphere with a radius of $\frac{1}{2}$ $\displaystyle \frac{1}{2}$ is cut out of a cube with a side length of $\displaystyle 2$. What is the volume of the resulting figure?

Possible Answers:

5.02 $\displaystyle 5.02$

6.67 $\displaystyle 6.67$

4.12 $\displaystyle 4.12$

7.48 $\displaystyle 7.48$

Correct answer:

7.48 $\displaystyle 7.48$

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.

$\text{Volume of Sphere}=\frac{4}{3}\pi(\frac{1}{2})^3=\frac{1}{6}\pi$ $\displaystyle \text{Volume of Sphere}=\frac{4}{3}\pi(\frac{1}{2})^3=\frac{1}{6}\pi$

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.

$\text{Volume of Figure}=8-\frac{1}{6}\pi=7.48$ $\displaystyle \text{Volume of Figure}=8-\frac{1}{6}\pi=7.48$

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

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Example Question #34 : How To Find The Volume Of A Cube

A sphere with a radius of $\displaystyle 4$ is cut out of a cube that has a side length of $\displaystyle 7$. What is the volume of the resulting figure?

Possible Answers:

98.09 $\displaystyle 98.09$

74.92 $\displaystyle 74.92$

101.45 $\displaystyle 101.45$

65.10 $\displaystyle 65.10$

Correct answer:

74.92 $\displaystyle 74.92$

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.

$\text{Volume of Sphere}=\frac{4}{3}\pi(4)^3=\frac{256}{3}\pi$ $\displaystyle \text{Volume of Sphere}=\frac{4}{3}\pi(4)^3=\frac{256}{3}\pi$

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.

$\text{Volume of Cube}=7^3=343$ $\displaystyle \text{Volume of Cube}=7^3=343$

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

$\text{Volume of Figure}=343-\frac{236}{3}\pi=74.92$ $\displaystyle \text{Volume of Figure}=343-\frac{236}{3}\pi=74.92$

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

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Intermediate Geometry : Cubes

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #25 : How To Find The Volume Of A Cube

Example Question #26 : How To Find The Volume Of A Cube

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

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

Example Question #29 : How To Find The Volume Of A Cube

Example Question #30 : How To Find The Volume Of A Cube

Example Question #31 : How To Find The Volume Of A Cube

Example Question #32 : How To Find The Volume Of A Cube

Example Question #33 : How To Find The Volume Of A Cube

Example Question #34 : How To Find The Volume Of A Cube

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