GMAT Math : Geometry

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Example Questions

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

$10 \sqrt{10}$

$40 \sqrt{10}$

$80 \sqrt{10}$

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 #111 : Rectangular Solids & Cylinders

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

Possible Answers:

$24 \sqrt{3}$

The correct answer is not given among the other responses.

$36 \sqrt{2}$

$12 \sqrt{6}$

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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Example Question #31 : Cubes

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

Possible Answers:

$27 \sqrt{3}$

$36 \sqrt{6}$

216

$72\sqrt{3}$

Correct answer:

216

Explanation:

The sidelength of the cube is the diameter of the inscribed sphere, which is twice that sphere's radius. 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 sidelength of the cube - is twice this, or 6. Cube this sidelength to get the volume of the cube:

$V = 6^{3} = 216$

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Example Question #591 : Geometry

The distance from one vertex of a cube to its opposite vertex is twelve feet. Give the volume of the cube.

Possible Answers:

$\frac{64 }{9} \textup{ yd}^{3}$

$32\sqrt{3} \textup{ yd}^{3}$

$32\sqrt{2} \textup{ yd}^{3}$

$\frac{64 \sqrt{3}}{9} \textup{ yd}^{3}$

$\frac{64 \sqrt{2}}{6} \textup{ yd}^{3}$

Correct answer:

$\frac{64 \sqrt{3}}{9} \textup{ yd}^{3}$

Explanation:

Since we are looking at yards, we will look at twelve feet as four yards.

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

$s^{2}+s^{2}+s^{2}=4^{2}$

$3s^{2}=16$

$s^{2}= \frac{16}{3}$

$s^{2}= \sqrt{\frac{16}{3}} = \frac{\sqrt{16}}{\sqrt{3}} = \frac{4}{\sqrt{3}}$ yards.

Cube this sidelength to get the volume:

$s^{3}= \left ( \frac{4}{\sqrt{3}} \right )^{3} = \frac{4^{3}}{(\sqrt{3})^{3}} = \frac{64 }{3\sqrt{3} }= \frac{64 \cdot \sqrt{3}}{3\sqrt{3}\cdot \sqrt{3} } = \frac{64 \sqrt{3}}{9}$ cubic yards.

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Example Question #32 : Cubes

An aquarium is shaped like a perfect cube; the area of each glass face is 6.25 square meters. If it is filled to the recommended $80\%$ capacity, then how much water will it contain?

Note: cubic meter = 1,000 liters.

Possible Answers:

The correct answer is not given among the other choices.

$10,000\textup{ L}$

$8,000\textup{ L}$

$12,500\textup{ L}$

$15,625\textup{ L}$

Correct answer:

$12,500\textup{ L}$

Explanation:

A perfect cube has square faces; if a face has area 6.25 square meters, then each side of each face measures the square root of this, or 2.5 meters. The volume of the tank is the cube of this, or

$2.5^{3} = 15.625$ cubic meters.

Its capacity in liters is $15.625 \times 1,000 = 15,625$ liters.

80% of this is

$15,625 \times 0.8 = 12,500$ liters.

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Example Question #112 : Rectangular Solids & Cylinders

Which choice comes closest to the volume of a cube with surface area 135,000 square centimeters?

Possible Answers:

$4.5 \textup{ m}^{3}$

$3 \textup{ m}^{3}$

$5 \textup{ m}^{3}$

$4 \textup{ m}^{3}$

$3.5 \textup{ m}^{3}$

Correct answer:

$3.5 \textup{ m}^{3}$

Explanation:

The suface area of a cube is six times the square of the length of one side, so solve for in the following:

$6s^{2} =135,000$

$s^{2} = 22,500$

s = 150

This is the sidelength in centimeters; since we are looking at meters, divide this by 100 to convert to $150 \div 100 = 1.5$ meters.

Cube this to get volume

$V =1.5^{3}= 3.375$ cubic meters.

Of the given choices, 3.5 cubic meters comes closest.

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Example Question #831 : Gmat Quantitative Reasoning

The distance from one vertex of a cube to its opposite vertex is one foot. Give the volume of the cube.

Possible Answers:

$192 \sqrt{3} \textup{ in}^{3}$

$266 \sqrt{2} \textup{ in}^{3}$

$216 \textup{ in}^{3}$

$288 \textup{ in}^{3}$

$64 \sqrt{3} \textup{ in}^{3}$

Correct answer:

$192 \sqrt{3} \textup{ in}^{3}$

Explanation:

Since we are looking at inches, we will look at one foot as twelve inches.

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

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

$3s^{2}=144$

$s^{2}=48$

$s= \sqrt{48} = 4\sqrt{3}$ inches.

Cube this sidelength to get the volume:

$s^{3}= \left ( 4\sqrt{3} \right )^{3} = 4^{3} \cdot \left ( \sqrt{3} \right ) ^{3} = 64 \cdot 3 \sqrt{3}= 192 \sqrt{3}$ cubic inches.

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Example Question #832 : Gmat Quantitative Reasoning

A cubic pool is usually filled at about $95\%$ of its total volume. If one side of the pool is m, how many liters of water will fill the pool to the desired capacity?

Possible Answers:

95,000

950,000

9,500,000

9,500

950

Correct answer:

950,000

Explanation:

To find the volume of a cube we use the formula:

V=s^3

In our case the total volume of the pool is 10³cubic meters.

Filling the pool to 95% of its total volume will require: $0.95\times1000 = 950$ cubic meters.

Now we need to convert from cubic meters to liters in order to answer the question.

Remember cubic meter = 1000 liters.

Thus,

$950 \times 1000=950,000$ .

Therefore, the number of liters of water needed is 950000 liters.

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Example Question #111 : Rectangular Solids & Cylinders

A sphere of surface area $324 \pi$ is inscribed inside a cube. Give the volume of the cube.

Possible Answers:

792

The correct answer is not given among the other responses.

5,832

648

Correct answer:

5,832

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 formula for the surface area of a sphere:

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

Substituting in the given value:

$4 \pi r^{2} = 324 \pi$

Dividing each side by $4\pi$ :

$r^{2} = \frac{324 \pi}{4 \pi }= 81$

Taking the square root of each side:

r = 9

The diameter is twice this, or . This is the length of the edge of the cube, so we can cube it to get the volume:

$V =18 ^{3}= 5,832$

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Example Question #833 : Gmat Quantitative Reasoning

A sphere with surface area $324 \pi$ circumscribes a cube. Give the volume of the cube.

Possible Answers:

5,832

648

792

The correct answer is not given among the other responses.

Correct answer:

The correct answer is not given among the other responses.

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 formula for the volume of a sphere, but first, we have to solve for the radius using the formula for the surface area of a sphere:

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

$4 \pi r^{2} = 324 \pi$

$r^{2} = \frac{324 \pi}{4 \pi }= 81$

r = 9

The diameter is twice this, or . This is the length of the diagonal of the cube. By the three-dimensional extension of the Pythagorean Theorem,

$s^{2}+ s^{2}+ s^{2} = 18^{2}$

$3s^{2} = 324$

$s^{2} =108$

$s = \sqrt{108} = \sqrt{36} \cdot \sqrt{3} = 6 \sqrt{3}$

The volume is the cube of this:

$V =(6 \sqrt{3}) ^{3}= 6^{3} \cdot \left (\sqrt{3} \right )^{3} = 216 \cdot 3 \cdot \sqrt{3} = 648 \sqrt{3}$ , which is not given as one of the choices.

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GMAT Math : Geometry

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #5 : Calculating The Volume Of A Cube

Example Question #111 : Rectangular Solids & Cylinders

Example Question #31 : Cubes

Example Question #591 : Geometry

Example Question #32 : Cubes

Example Question #112 : Rectangular Solids & Cylinders

Example Question #831 : Gmat Quantitative Reasoning

Example Question #832 : Gmat Quantitative Reasoning

Example Question #111 : Rectangular Solids & Cylinders

Example Question #833 : Gmat Quantitative Reasoning

All GMAT Math Resources

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