GMAT Math : Cubes

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Example Question #5 : Calculating The Diagonal Of A Cube

A given cube has an edge length of . What is the length of the diagonal of the cube?

Possible Answers:

$18\sqrt{2}$

$9\sqrt{3}$

$9\sqrt{2}$

Not enough information provided

$18\sqrt{3}$

Correct answer:

$9\sqrt{3}$

Explanation:

The diagonal of a cube with an edge length can be defined by the equation $d=s\sqrt{3}$ . Given s=9 in this instance, $d=9\sqrt{3}$ .

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Example Question #6 : Calculating The Diagonal Of A Cube

A given cube has an edge length of $\sqrt{3}$ . What is the length of the diagonal of the cube?

Possible Answers:

$3\sqrt{3}$

None of the above

$2\sqrt{3}$

Correct answer:

Explanation:

The diagonal of a cube with an edge length can be defined by the equation $d=s\sqrt{3}$ . Given $s=\sqrt{3}$ in this instance, $d=\sqrt{3}\left ( \sqrt{3} \right )=3$ .

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Example Question #7 : Calculating The Diagonal Of A Cube

If a cube has a side length of , what is the length of its diagonal?

Possible Answers:

$3\sqrt{2}$

$3\sqrt{4}$

$4\sqrt{2}$

$2\sqrt{3}$

$4\sqrt{3}$

Correct answer:

$4\sqrt{3}$

Explanation:

The diagonal of a cube is the hypotenuse of a right triangle whose height is one side and whose base is the diagonal of one of the faces. First we must use the Pythagorean theorem to find the length of the diagonal of one of the faces, and then we use the theorem again with this value and length of one side of the cube to find the length of its diagonal:

a^2+b^2=c^2

4^2+4^2=c^2

$c^2=32\rightarrow c=\sqrt{32}$

So this is the length of the diagonal of one of the faces, which we plug into the Pythagorean theorem with the length of one side to find the length of the diagonal for the cube:

h^2+c^2=d^2

$4^2+(\sqrt{32})^2=d^2$

$d^2=48\rightarrow d=\sqrt{48}=\sqrt{3*16}=4\sqrt{3}$

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Example Question #1 : Calculating The Surface Area Of A Cube

What is the surface area of a box that is 3 feet long, 2 feet wide, and 4 feet high?

Possible Answers:

$\dpi{100} \small 60$

$\dpi{100} \small 24$

$\dpi{100} \small 52$

$\dpi{100} \small 40$

$\dpi{100} \small 43$

Correct answer:

$\dpi{100} \small 52$

Explanation:

$\dpi{100} \small SA = 2lw + 2lh + 2wh = 2\times 3\times 2 + 2 \times 3\times 4 + 2\times 2\times 4 = 52$

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Example Question #2 : Calculating The Surface Area Of A Cube

What is the surface area of a cube with side length 4?

Possible Answers:

384

Correct answer:

Explanation:

$SA=6\ast side^{2}=6\ast 4^{2}=6\ast 16=96$

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Example Question #811 : Gmat Quantitative Reasoning

The surface area of a certain cube is 150 square feet. If the width of the cube is increased by 2 feet, the length decreased by 2 feet and the height increased by 1 foot, what is the new surface area?

Possible Answers:

$151\ ft^{2}$

$192\ ft^{2}$

$252\ ft^{2}$

$162\ ft^{2}$

$243\ ft^{2}$

Correct answer:

$162\ ft^{2}$

Explanation:

The first step to answering this qestion is to determine the original length of the sides of the cube. The surface area of a cube is given by:

$S_{A} = 6x^{2}$

Where is the length of each side. This tells us that for our cube:

$150 = 6x^{2}$ ==> $25 = x^{2}$ ==> x = 5

If the width increases by 2, the length decreases by 2 and the height increases by 1:

w=7 , l=3 , h=6

We now have a rectangular prism. The surface area of a rectangular prism is given by:

$S_{A} = 2(wl + lh + hw)$

For our prism:

$2((7\times 3) + (7 \times 6) + (6 \times 3)) = 2(21 + 42 + 18) = 162$

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Example Question #3 : Calculating The Surface Area Of A Cube

What is the surface area of a cube with a side length of ?

Possible Answers:

Correct answer:

Explanation:

SA=6s^2

SA=6(2)^2

SA=6*4

SA=24

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Example Question #5 : Calculating The Surface Area Of A Cube

A cube is inscribed inside a sphere with surface area $100 \pi$ . Give the volume of the cube.

Possible Answers:

$250 \sqrt{2}$

$\frac{1,000 \sqrt{2} } {6 }$

500

$\frac{1,000 \sqrt{3} } {9 }$

Correct answer:

$\frac{1,000 \sqrt{3} } {9 }$

Explanation:

Each diagonal of the inscribed cube is a diameter of the sphere, so its length is the sphere's diameter, or twice its radius.

The sphere has surface area $80 \pi$ , so the radius is calculated as follows:

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

$4 \pi r^{2} = 100 \pi$

$r^{2} =100 \pi \div 4 \pi = 25$

r = 5

The diameter of the circle - and the length of a diagonal of the cube - is twice this, or 10.

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

$s^{2}+s^{2}+s^{2}=10^{2}$

$3s^{2}=100$

$s^{2}=\frac{100}{3}$

$s = \sqrt{\frac{100}{3}} = \frac{\sqrt{100}}{\sqrt{3}} = \frac{10}{\sqrt{3}}$

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

$V=\left ( \frac{10}{\sqrt{3}} \right )^{3} = \frac{10^{3} } {\left (\sqrt{3} \right )^{3} } =\frac{1,000 } {3 \sqrt{3} } =\frac{1,000 \cdot \sqrt{3} } {3 \sqrt{3} \cdot \sqrt{3} } =\frac{1,000 \sqrt{3} } {9 }$

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Example Question #6 : Calculating The Surface Area Of A Cube

A sphere of volume $28 8 \pi$ is inscribed inside a cube. Give the surface area of the cube.

Possible Answers:

216

144

288

864

Correct answer:

864

Explanation:

The diameter of a sphere is equal to the length of an edge of the cube in which it is inscribed. We can derive the radius using the volume formula:

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

$288 \pi= \frac{4}{3} \pi r^{3}$

$\frac{4}{3} r^{3} = 288$

$\frac{3}{4} \cdot \frac{4}{3} r^{3} = \frac{3}{4} \cdot 288$

$r^{3} = 216$

r = 6

Twice this, or 12, is the diameter, and, subsequently, the length of an edge of the cube. If s = 12 , the surface area is

$A = 6s^{2} = 6 \cdot 12^{2}= 6 \cdot 144 = 864$

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Example Question #7 : Calculating The Surface Area Of A Cube

A cube is inscribed inside a sphere of volume $2 88 \pi$ . Give the surface area of the cube.

Possible Answers:

288

216

144

864

Correct answer:

288

Explanation:

The diameter of a sphere is equal to the length of a diagonal of the cube it circumscribes. We can derive the radius using the volume formula:

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

$288 \pi= \frac{4}{3} \pi r^{3}$

$\frac{4}{3} r^{3} = 288$

$\frac{3}{4} \cdot \frac{4}{3} r^{3} = \frac{3}{4} \cdot 288$

$r^{3} = 216$

r = 6

Twice this, or 12, is the diameter, and, subsequently, the length of a diagonal of the cube. By an extension of the Pythagorean Theorem, if is the length of an edge of the cube,

$s^{2}+ s^{2}+ s^{2} = 12^{2}$

$3s^{2} = 144$

$s^{2} = 48$

The surface area is six times this:

$SA = 6s^{2} = 6 \cdot 48 = 288$

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GMAT Math : Cubes

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #5 : Calculating The Diagonal Of A Cube

Example Question #6 : Calculating The Diagonal Of A Cube

Example Question #7 : Calculating The Diagonal Of A Cube

Example Question #1 : Calculating The Surface Area Of A Cube

Example Question #2 : Calculating The Surface Area Of A Cube

Example Question #811 : Gmat Quantitative Reasoning

Example Question #3 : Calculating The Surface Area Of A Cube

Example Question #5 : Calculating The Surface Area Of A Cube

Example Question #6 : Calculating The Surface Area Of A Cube

Example Question #7 : Calculating The Surface Area Of A Cube

All GMAT Math Resources

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