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SSAT Upper Level Math : Geometry

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Example Questions

Example Question #1328 : Concepts

A cube has six square faces, each with area 64 square inches. Using the conversion factor 1 inch = 2.5 centimeters, give the volume of this cube in cubic centimeters, rounding to the nearest whole number.

Possible Answers:

800 cubic centimeters

8,000 cubic centimeters

400 cubic centimeters

4,000 cubic centimeters

2,000 cubic centimeters

Correct answer:

8,000 cubic centimeters

Explanation:

The volume of a cube is the cube of its sidelength, which is also the sidelength of each square face. This sidelength is the square root of the area 64:

$\sqrt{64}=8$ inches.

Multiply this by 2.5 to get the sidelength in centimeters:

$\times 2.5$ = 20 centimeters.

The cube of this is

$20^{3}$ =8000 cubic centimeters

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Example Question #1329 : Concepts

A cube has a side length of 5 inches. Give the volume and surface area of the cube.

Possible Answers:

25 in^3 ,150 in^2

25 in^3 ,75 in^2

125 in^3 ,150 in^2

125 in^3 ,75 in^2

250in^3, 150 in^2

Correct answer:

125 in^3 ,150 in^2

Explanation:

A cube has all edges the same length. The volume of a cube is found by multiplying the length of any edge by itself twice. As a formula:

Volume=s^3 where is the length of any edge of the cube.

The Surface Area of a cube can be calculated as 6s^2 .

So we get:

Volume =s^3=5^3=125 in^3

Surface area $=6s^2=6\times 5^2=6\times 25=150 in^2$

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Example Question #2 : How To Find The Volume Of A Figure

A cheese seller has a 2 foot x 2 foot x 2 foot block of gouda and she wants to cut it into smaller gouda cubes that are 1.5 inches on a side. How many cubes can she cut?

Possible Answers:

4096

2048

512

256

1024

Correct answer:

4096

Explanation:

First we need to determine how many of the small cubes of gouda would fit along one dimension of the large cheese block. One edge of the large block is 24 inches, so 16 smaller cubes $(24\div 1.5)$ would fit along the edge. Now we simply cube this one dimension to see how many cubes fit within the whole cube. $16^{3}=4096$ .

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Example Question #4 : How To Find The Volume Of A Cube

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

Possible Answers:

$\frac{N^{3} \sqrt{2} }{4}$

$\frac{N^{3} }{3 }$

$\frac{N^{3} }{2 }$

$3N\sqrt{N}$

$\frac{N^{3} \sqrt{3} }{9 }$

Correct answer:

$\frac{N^{3} \sqrt{3} }{9 }$

Explanation:

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

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

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

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

$s =\sqrt{\frac{N^{2}}{3}} = \frac{N}{\sqrt{3}} = \frac{N\sqrt{3}}{3}$

Cube this sidelength to get the volume:

$s^{3} = \left ( \frac{N\sqrt{3}}{3} \right )^{3} = \frac{N^{3} \left ( \sqrt{3}\right ) ^{3} }{3^{3} } = \frac{N^{3} \left (3 \sqrt{3}\right ) }{27 }= \frac{N^{3} \sqrt{3} }{9 }$

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Example Question #631 : Geometry

Give the volume of a cube with surface area .

Possible Answers:

$6\sqrt[3]{N^{2}}$

$\frac{N\sqrt{ 6N} }{36}$

$\frac{N^{3}}{216}$

$\frac{N^{3}}{36}$

$6N\sqrt{ 6N}$

Correct answer:

$\frac{N\sqrt{ 6N} }{36}$

Explanation:

Let be the length of one edge of the cube. Since its surface area is , one face has one-sixth of this area, or $\frac{N}{6}$ . Therefore,

$s^{2} = \frac{N}{6}$ ,

and

$s =\sqrt{ \frac{N}{6}}= \frac{ \sqrt{ N}}{\sqrt{ 6}}= \frac{ \sqrt{ 6N}}{6}$

Cube this sidelength to get the volume:

$s ^{3}= \left ( \frac{ \sqrt{ 6N}}{6} \right )^{3} = \frac{\left ( \sqrt{ 6N} \right )^{3}}{6^{3}}= \frac{ 6N\sqrt{ 6N} }{216}= \frac{N\sqrt{ 6N} }{36}$

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Example Question #5 : How To Find The Volume Of A Cube

The length of a diagonal of one face of a cube is 10. Give the volume of the cube.

Possible Answers:

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

$500\sqrt{2}$

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

$250\sqrt{2}$

$125\sqrt{2}$

Correct answer:

$250\sqrt{2}$

Explanation:

Since a diagonal of a square face of the cube is 10, each side of each square has length $\frac{\sqrt{2}}{2}$ times this, or $10 \cdot \frac{\sqrt{2}}{2} = 5\sqrt{2}$ .

Cube this to get the volume of the cube:

$\left (5\sqrt{2} \right )^{3}= 5^{3} \left ( \sqrt{2} \right )^{3} = 125 \left (2 \sqrt{2} \right ) = 250 \sqrt{2}$ .

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Example Question #3 : How To Find The Volume Of A Figure

An aquarium is shaped like a perfect cube; the area of each glass face is 1.44 square meters. If it is filled to the recommended 90% capacity, then, to the nearest hundred liters, how much water will it contain?

Note: 1 cubic meter = 1,000 liters.

Possible Answers:

$1,400\textup{ L}$

$1,700\textup{ L}$

$1,200\textup{ L}$

$1,600 \textup{ L}$

$1,300\textup{ L}$

Correct answer:

$1,600 \textup{ L}$

Explanation:

A perfect cube has square faces; if a face has area 1.44 square meters, then each side of each face measures the square root of this, or 1.2 meters. The volume of the tank is the cube of this, or

$1.2^{3} = 1.728$ cubic meters.

Its capacity in liters is $1.728 \times 1,000 = 1,728$ liters.

90% of this is

$1,728 \times 0.9 = 1,555.2$ liters.

This rounds to 1,600 liters, the correct response.

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Example Question #21 : Volume Of A Three Dimensional Figure

Give the volume of a cube with surface area 3 square meters.

Possible Answers:

$125,000 \textup{ cm}^{3}$

$250,000 \textup{ cm}^{3}$

$250,000\sqrt{2} \textup{ cm}^{3}$

$125,000\sqrt{2} \textup{ cm}^{3}$

$125,000\sqrt{3} \textup{ cm}^{3}$

Correct answer:

$250,000\sqrt{2} \textup{ cm}^{3}$

Explanation:

Let be the length of one edge of the cube. Since its surface area is 3 square meters, one face has one-sixth of this area, or $\frac{3}{6} = \frac{1}{2}$ square meters. Therefore, $s^{2} = \frac{1}{2}$ , and $s = \sqrt{ \frac{1}{2}} = \frac{\sqrt{1}}{\sqrt{2}}=\frac{1 \cdot \sqrt{2}}{\sqrt{2}\cdot \sqrt{2}}= \frac{\sqrt{2}}{2}$ meters.

The choices are in centimeters, so multiply this by 100 - the sidelength is

$\frac{\sqrt{2}}{2} \cdot 100 = 50 \sqrt{2}$ centimeters.

The volume is the cube of this, or $V= \left ( 50 \sqrt{2} \right )^{3}= 50^{3} \cdot \left ( \sqrt{2} \right )^{3}= 125,000 \cdot 2 \sqrt{2}= 250,000\sqrt{2}$ cubic centimeters.

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Example Question #7 : How To Find The Volume Of A Cube

The length of a diagonal of a cube is $15\sqrt{2}$ . Give the volume of the cube.

Possible Answers:

$750 \sqrt{2}$

$750 \sqrt{6}$

$1,500 \sqrt{2}$

$750 \sqrt{3}$

$1,500 \sqrt{3}$

Correct answer:

$750 \sqrt{6}$

Explanation:

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

$s^{2}+s^{2}+s^{2}= (15 \sqrt{2})^{2}$

$3s^{2}=15 ^{2}\cdot \left ( \sqrt{2} \right )^{2}$

$3s^{2}=225 \cdot 2 = 450$

$s^{2} = 150$

$s = \sqrt{150 } = \sqrt{25} \cdot \sqrt{6} = 5\sqrt{6}$

Cube the sidelength to get the volume:

$s ^{3 }= \left ( 5\sqrt{6} \right )^{3 }$

$s ^{3 }= 5^{3 } \left (\sqrt{6} \right )^{3 } = 125 \cdot 6 \sqrt{6} = 750 \sqrt{6}$

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Example Question #632 : Geometry

The length of a diagonal of one face of a cube is $16\sqrt{2}$ . Give the volume of the cube.

Possible Answers:

$8,192\sqrt{2}$

4,096

32,768

$65,536\sqrt{2}$

The correct answer is not among the other responses.

Correct answer:

4,096

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 $16 \sqrt{2} \div \sqrt{2}= 16$ . Cube this to get the volume:

$16^{3} = 4,096$

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SSAT Upper Level Math : Geometry

Study concepts, example questions & explanations for SSAT Upper Level Math

All SSAT Upper Level Math Resources

Example Questions

Example Question #1328 : Concepts

Example Question #1329 : Concepts

Example Question #2 : How To Find The Volume Of A Figure

Example Question #4 : How To Find The Volume Of A Cube

Example Question #631 : Geometry

Example Question #5 : How To Find The Volume Of A Cube

Example Question #3 : How To Find The Volume Of A Figure

Example Question #21 : Volume Of A Three Dimensional Figure

Example Question #7 : How To Find The Volume Of A Cube

Example Question #632 : Geometry

All SSAT Upper Level Math Resources

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