GMAT Math : Cubes

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Example Question #8 : Calculating The Surface Area Of A Cube

Cube A is inscribed inside a sphere, which is inscribed inside Cube B. Give the ratio of the surface area of Cube B to that of Cube A.

Possible Answers:

2:1

3:2

4:1

3:1

9:4

Correct answer:

3:1

Explanation:

Suppose the sphere has diameter .

Then Cube B, the circumscribing cube, has as its edge length the diameter , and its surface area is $6d^{2}$ .

Also, Cube A, the inscribed cube, has this diameter as the length of its diagonal. If is the length of an edge, then from the three-dimensional extension of the Pythagorean Theorem,

$s^{2}+ s^{2}+ s^{2} = d^{2}$

$3s^{2} = d^{2}$

The surface area is $SA = 6s^{2}$ , so

$SA = 6s ^{2 } = 2 \cdot 3s^{2} = 2 d^{2}$ .

The ratio of the surface areas is

$\frac{6d^{2}}{ 2 d^{2}} = \frac{3}{1}$

The correct choice is 3:1 .

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Example Question #9 : Calculating The Surface Area Of A Cube

The length of one side of a cube is 4 meters. What is the surface area of the cube?

Possible Answers:

m^2

128 m^2

150 m^2

m^2

Correct answer:

m^2

Explanation:

By definition, all sides of a cube are equal in length, so each face is a square, There are six faces on a cube, so its total surface area is six times the area of one of its square faces. If one of its sides is 4 meters, then this will also be the other dimension of one of its square faces, so the total surface area is:

SA=6L^2=6(4)^2=96 m^2

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Example Question #4 : Calculating The Surface Area Of A Cube

Find the surface area of a cube whose side length is .

Possible Answers:

Correct answer:

Explanation:

To solve, remember that the equation for surface area of a cube is:

SA=6s^2=6(4^2)=6*16=96

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Example Question #101 : Rectangular Solids & Cylinders

Aperture labs makes a variety of cubes. If each cube has a volume of , what is the surface area of the cube?

Possible Answers:

45.67ft^2

1.82ft^2

3.30ft^2

19.81ft^2

Correct answer:

19.81ft^2

Explanation:

Aperture labs makes a variety of cubes. If each cube has a volume of , what is the surface area of the cube?

Let's work backwards from our goal in this question.

We know that we need to find surface area. To find surface area of a cube, we can use the following equation:

SA=6l^2

Where l is the length of one side.

Next, let's look at the volume formula:

$V_{cube}=l^3$

So, we can find our length

6ft^3=l^3

$l=(6)^{\frac{1}{3}}$

Let's leave l like that for the moment, and use it to find our surface area.

$SA=6((6)^{\frac{1}{3}})^2\approx19.81ft^2$

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Example Question #1 : Calculating The Volume Of A Cube

What is the volume of a cube with a side length of ?

Possible Answers:

Correct answer:

Explanation:

V=s^3

V=2^3

V=8

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Example Question #2 : Calculating The Volume Of A Cube

The length, width, and height of a rectangular prism, in inches, are three different prime numbers. All three dimensions are between six feet and seven feet. What is the volume of the prism?

Possible Answers:

$490,779\textrm{ in}^{3}$

It is impossible to tell from the information given.

$504,889\textrm{ in}^{3}$

$444,059\textrm{ in}^{3}$

$478,661 \textrm{ in}^{3}$

Correct answer:

$478,661 \textrm{ in}^{3}$

Explanation:

Six feet and seven feet are equal to, respectively, 72 inches and 84 inches. There are three different prime numbers between 72 and 84 - 73, 79, and 83 - so these are the three dimensions of the prism in inches. The volume of the prism is

$73 \times 79 \times 83 = 478,661$ cubic inches.

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Example Question #3 : Calculating The Volume Of A Cube

The length of a diagonal of one face of a cube is $5\sqrt{2}$ . Give the volume of the cube.

Possible Answers:

The correct answer is not among the other responses.

$125 \sqrt{2}$

125

$\frac{125 \sqrt{2}}{2}$

250

Correct answer:

125

Explanation:

A diagonal of a square has length $\sqrt{2}$ times that of a side, so each side of each square face of the cube has length $5 \sqrt{2} \div \sqrt{2}= 5$ . Cube this to get the volume:

$V= 5^{3} = 125$

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Example Question #4 : Calculating The Volume Of A Cube

The length of a diagonal of a cube is $\sqrt{17}$ . Give the volume of the cube.

Possible Answers:

$\frac{ 17 \sqrt{ 51} }{ 3} \right )$

$\frac{ 17 \sqrt{ 51} }{ 9} \right )$

$\frac{ 17 \sqrt{ 34} }{ 6} \right )$

289

$\frac{ 17 \sqrt{ 34} }{ 2} \right )$

Correct answer:

$\frac{ 17 \sqrt{ 51} }{ 9} \right )$

Explanation:

Let be the length of one edge of the cube. By the three-dimensional extension of the Pythagorean Theorem,

$s^{2}+s^{2}+s^{2}= ( \sqrt{17})^{2}$

$3s^{2}= 17$

$s^{2}= \frac{17}{3}$

$s =\sqrt{ \frac{17}{3}}$

Cube the sidelength to get the volume:

$s^{3} =\left (\sqrt{ \frac{17}{3}} \right )^{3}$

$=\frac{ \left (\sqrt{ 17} \right )^{3}}{\left (\sqrt{3} \right )^{3}} \right )$

$=\frac{ 17 \sqrt{ 17} }{ 3 \sqrt{3} } \right )$

$=\frac{ 17 \sqrt{ 17} \cdot \sqrt{3}}{ 3 \sqrt{3} \cdot \sqrt{3}} \right )$

$=\frac{ 17 \sqrt{ 51} }{ 9} \right )$

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Example Question #5 : Calculating The Volume Of A Cube

A sphere with surface area $40 \pi$ is inscribed inside a cube. Give the volume of the cube.

Possible Answers:

8 0

$80 \sqrt{10}$

$10 \sqrt{10}$

$40 \sqrt{10}$

Correct answer:

$80 \sqrt{10}$

Explanation:

The sidelength of the cube is the diameter of the inscribed sphere, which is twice that sphere's radius. The sphere has surface area $40 \pi$ , so the radius is calculated as follows:

$4 \pi r^{2} = SA$

$4 \pi r^{2} = 40 \pi$

$r^{2} = 40 \pi \div 4 \pi = 10$

$r = \sqrt{10}$

The diameter of the sphere - and the sidelength of the cube - is twice this, or $2 \sqrt{10}$ .

Cube this sidelength to get the volume of the cube:

$V= \left ( 2 \sqrt{10} \right )^{3} = 2^{3} \cdot\left ( \sqrt{10 } \right )^{3} = 8 \cdot 10 \sqrt{10} = 80 \sqrt{10}$

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Example Question #6 : Calculating The Volume Of A Cube

A cube is inscribed inside a sphere with volume $36 \pi$ . Give the volume of the cube.

Possible Answers:

$36 \sqrt{2}$

$24 \sqrt{3}$

$12 \sqrt{6}$

The correct answer is not given among the other responses.

Correct answer:

$24 \sqrt{3}$

Explanation:

The diameter of the circle - twice its radius - coincides with the length of a diagonal of the inscribed cube. The sphere has volume $36 \pi$ , so the radius is calculated as follows:

$\frac{4}{3} \pi r^{3} = V$

$\frac{4}{3} \pi r^{3} = 36 \pi$

$r^{3} = 36\pi \div \left (\frac{4}{3} \pi \right )$

$r^{3} = 27$

$r = \sqrt[3]{27} = 3$

The diameter of the sphere - and the length of a diagonal of the cube - is twice this, or 6.

Now, let be the length of one edge of the cube. By the three-dimensional extension of the Pythagorean Theorem,

$s^{2}+s^{2}+s^{2}=6^{2}$

$3s^{2}=36$

$s^{2}=12$

$s = \sqrt{12} = \sqrt{4} \cdot \sqrt{3} = 2 \sqrt{3}$

The volume of the cube is the cube of this, or

$V=\left (2 \sqrt{3} \right )^{3} =2^{3} \left ( \sqrt{3} \right )^{3} = 8 \cdot 3 \sqrt{3} = 24 \sqrt{3}$

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GMAT Math : Cubes

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #8 : Calculating The Surface Area Of A Cube

Example Question #9 : Calculating The Surface Area Of A Cube

Example Question #4 : Calculating The Surface Area Of A Cube

Example Question #101 : Rectangular Solids & Cylinders

Example Question #1 : Calculating The Volume Of A Cube

Example Question #2 : Calculating The Volume Of A Cube

Example Question #3 : Calculating The Volume Of A Cube

Example Question #4 : Calculating The Volume Of A Cube

Example Question #5 : Calculating The Volume Of A Cube

Example Question #6 : Calculating The Volume Of A Cube

All GMAT Math Resources

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