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Intermediate Geometry : How to find the volume of a cube

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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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Example Question #71 : Cubes

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

Possible Answers:

25.76

23.01

26.92

24.85

Correct answer:

25.76

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.

$\text{Volume of Sphere}=\frac{4}{3}\pi(\frac{2}{3})^3=\frac{32}{81}\pi$

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.

$\text{Volume of Cube}=3^3=27$

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

$\text{Volume of Figure}=27-\frac{32}{81}\pi=25.76$

Make sure to round to places after the decimal.

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Example Question #72 : Cubes

Pyramid

The above figure shows a triangular pyramid inscribed inside a cube. The pyramid has volume 100. Give the volume of the cube.

Possible Answers:

$666\frac{2}{3}$

400

750

800

600

Correct answer:

600

Explanation:

Let be the length of one side of the cube. Then the base of the pyramid is a right triangle with legs of length . The area of a right triangle is half the product of the lengths of the legs, so the base has area

$B = \frac{1}{2} s \cdot s = \frac{1}{2} s ^{2}$

The volume of a pyramid is one third the product of its height and the area of its base. The height of the pyramid is , so the volume is

$V_{1} = \frac{1}{3} Bh = \frac{1}{3} \cdot \frac{1}{2} s^{2} \cdot s = \frac{1}{6} s^{3}$

The volume of the cube is $V_{2} = s^{3}$ , so

$V_{1} = \frac{1}{6} V_{2}$ ,

or, equivalently,

$6 \cdot V_{1} = 6 \cdot \frac{1}{6} V_{2}$

$V_{2}= 6 \cdot V_{1}$

That is, the volume of the cube is six times that of the pyramid, and, since the pyramid has volume 100, the volume of the cube is $6 \cdot 100 =600$ .

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Intermediate Geometry : How to find the volume of a cube

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

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

Example Question #71 : Cubes

Example Question #72 : Cubes

All Intermediate Geometry Resources

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