GMAT Math : Calculating the surface area of a cube

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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:

$\dpi{100} \small 24$

$\dpi{100} \small 43$

$\dpi{100} \small 52$

$\dpi{100} \small 60$

$\dpi{100} \small 40$

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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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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GMAT Math : Calculating the surface area of a cube

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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

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

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