Calculating the surface area of a cube - GMAT Quantitative Reasoning

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Question

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

Answer

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

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Question

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

Answer

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

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Question

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?

Answer

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:

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

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

, ,

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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Question

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

Answer

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Question

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

Answer

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:

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

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

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

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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Question

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

Answer

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:

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

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

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

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

$r^{3} = 216$

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

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

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Question

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

Answer

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:

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

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

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

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

$r^{3} = 216$

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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Question

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.

Answer

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 .

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Question

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

Answer

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:

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Question

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

Answer

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

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Question

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

Answer

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:

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

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

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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