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Intermediate Geometry : Solid Geometry

Study concepts, example questions & explanations for Intermediate Geometry

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Example Questions

Example Question #61 : Cubes

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

2765

2995

2835

Correct answer:

2835

Explanation:

Start by finding the volume of the cube.

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

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

Next, find the volume of the rectangular prism.

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

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

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

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

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

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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:

880

1200

1140

990

Correct answer:

1140

Explanation:

Start by finding the volume of the cube.

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

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

Next, find the volume of the rectangular prism.

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

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

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

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

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

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Example Question #62 : Cubes

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:

1020

960

810

1420

Correct answer:

960

Explanation:

Start by finding the volume of the cube.

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

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

Next, find the volume of the rectangular prism.

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

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

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

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

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

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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

2408

2296

2222

Correct answer:

2304

Explanation:

Start by finding the volume of the cube.

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

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

Next, find the volume of the rectangular prism.

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

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

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

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

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

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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

17750

19005

18920

Correct answer:

20250

Explanation:

Start by finding the volume of the cube.

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

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

Next, find the volume of the rectangular prism.

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

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

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

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

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

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

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

Possible Answers:

25.10

34.11

26.66

30.49

Correct answer:

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.

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

Plug in the given radius to find the volume.

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

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

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

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

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

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

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

Make sure to round to 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 is cut out of a cube that has a side length of . What is the volume of the resulting figure?

Possible Answers:

109.58

120.81

106.65

122.12

Correct answer:

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.

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

Plug in the given radius to find the volume.

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

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

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

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

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

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

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

Make sure to round to 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 is cut out of a cube with a side length of . What is the volume of the resulting figure?

Possible Answers:

102.90

110.51

101.21

98.20

Correct answer:

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.

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

Plug in the given radius to find the volume.

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

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

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

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

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

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

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

Make sure to round to 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 $\frac{1}{2}$ is cut out of a cube with a side length of . What is the volume of the resulting figure?

Possible Answers:

4.12

7.48

5.02

6.67

Correct answer:

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.

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

Plug in the given radius to find the volume.

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

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

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

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

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

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

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

Make sure to round to 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 is cut out of a cube that has a side length of . What is the volume of the resulting figure?

Possible Answers:

65.10

74.92

98.09

101.45

Correct answer:

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.

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

Plug in the given radius to find the volume.

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

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

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

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

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

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

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

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 #61 : Cubes

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

Example Question #62 : Cubes

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

All Intermediate Geometry Resources

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