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ISEE Upper Level Math : Solid Geometry

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Example Questions

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Example Question #1 : Cubes

The distance from one vertex of a perfectly cubic aquarium to its opposite vertex is 1.5 meters. Give the volume of the aquarium in liters.

1 cubic meter = 1,000 liters.

Possible Answers:

The correct answer is not given among the other responses.

$843\frac{3}{4} \textup{ L}$

$421\frac{7}{8} \textup{ L}$

$750\textup{ L}$

$375 \textup{ L}$

Correct answer:

The correct answer is not given among the other responses.

Explanation:

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

$s^{2}+s^{2}+s^{2}=1.5^{2}$

$3s^{2}=2.25$

$s^{2} = 0.75 = \frac{3}{4}$

$s = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{\sqrt{4}}} = \frac{\sqrt{3}}{2}$ meters.

Cube this sidelength to get the volume:

$s ^{3 } =\left ( \frac{\sqrt{3}}{2} \right )^{3 }= \frac{\left (\sqrt{3}\right )^{3 }}{2^{3 } } = \frac{ 3\sqrt{3} }{8 }$ cubic meters.

To convert this to liters, multiply by 1,000:

$\frac{ 3\sqrt{3} }{8 } \cdot 1,000 = \frac{ 3\sqrt{3} }{8 } \cdot \frac{1,000}{1} = \frac{ 3\sqrt{3} }{1 } \cdot \frac{125}{1}= 375\sqrt{3}$ liters.

This is not among the given responses.

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Example Question #4 : Finding Volume Of A Cube

Give the volume of a cube with surface area 240 square inches.

Possible Answers:

$40 \sqrt{10} \textup{ in}^{3}$

$120 \textup{ in}^{3}$

$80 \textup{ in}^{3}$

$80 \sqrt{10} \textup{ in}^{3}$

$120 \sqrt{10} \textup{ in}^{3}$

Correct answer:

$80 \sqrt{10} \textup{ in}^{3}$

Explanation:

Let be the length of one edge of the cube. Since its surface area is 240 square inches, one face has one-sixth of this area, or $\frac{240}{6} = 40$ square inches. Therefore, $s^{2} = 40$ , and $s = \sqrt{40} = 2\sqrt{10}$ .

The volume is the cube of this, or $V= \left ( 2\sqrt{10} \right )^{3}= 2^{3} \cdot \left ( \sqrt{10} \right )^{3}= 8 \cdot 10 \sqrt{10}= 80 \sqrt{10}$ cubic inches.

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Example Question #1 : Finding Volume Of A Rectangular Prism

An aquarium is shaped like a perfect cube; the perimeter of each glass face is 8.4 meters. If it is filled to the recommended $90\%$ capacity, then, to the nearest hundred cubic liters, how much water will it contain?

Possible Answers:

$4\textup,000\textup{ L}$

$8\textup,300\textup{ L}$

Insufficient information is given to answer the question.

$9\textup,300\textup{ L}$

$4\textup,400\textup{ L}$

Note: $1\textup{ cubic meter}= 1\textup{,}000\textup{ cubic liters}$

Correct answer:

$8\textup,300\textup{ L}$

Explanation:

A perfect cube has square faces; if a face has perimeter 8.4 meters, then each side of each face measures one fourth of this, or 2.1 meters. The volume of the tank is the cube of this, or

$2.1^{3} = 9.261$ cubic meters.

Its capacity in liters is $9.261 \times 1,000 = 9,261$ liters.

$90\%$ of this is

$9,261 \times 0.9 = 8,335$ liters.

This rounds to $8\textup,300$ liters, the correct response.

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Example Question #1 : How To Find The Volume Of A Cube

Your friend gives you a puzzle cube for your birthday. If the length of one edge is 5cm, what is the volume of the cube?

Possible Answers:

625cm^3

25cm^3

125cm^3

15cm^3

Correct answer:

125cm^3

Explanation:

Your friend gives you a puzzle cube for your birthday. If the length of one edge is 5cm, what is the volume of the cube?

To find the volume of a cube, use the following formula:

$V_{cube}=s^3$

Where s is the side length.

Plug in what we know to get our answer:

V=(5cm)^3=125cm^3

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

A cube has a side length of 9 mm , what is the volume of the cube?

Possible Answers:

81mm^3

729mm^3

486mm^3

72.9mm^3

Correct answer:

729mm^3

Explanation:

A cube has a side length of 9 mm , what is the volume of the cube?

To find the volume of a cube, use the following formula:

$V_{cube}=s^3$

Plug in our known side length and solve

$V_{cube}=(9mm)^3=729mm^3$

Making our answer:

729mm^3

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Example Question #11 : How To Find The Volume Of A Cube

One of your holiday gifts is wrapped in a cube-shaped box.

If one of the edges has a length of 6 inches, what is the volume of the box?

Possible Answers:

96in^2

216in^2

216in^3

144in^3

Correct answer:

216in^3

Explanation:

One of your holiday gifts is wrapped in a cube-shaped box.

If one of the edges has a length of 6 inches, what is the volume of the box?

Find the volume of a cube via the following:

$V_{cube}=s^3=(6in)^3=216in^3$

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Example Question #12 : Cubes

Find the volume of a cube with a height of 3in.

Possible Answers:

$9\text{in}^3$

$18\text{in}^3$

$12\text{in}^3$

$27\text{in}^3$

$\text{There is not enough information to solve the problem.}$

Correct answer:

$27\text{in}^3$

Explanation:

To find the volume of a cube, we will use the following formula:

V = a^3

where a is the length of any side of the cube.

Now, we know the height of the cube is 3in. Because it is a cube, all sides (lengths, widths, height) are the same. That is why we can find any length for the formula.

Knowing this, we can substitute into the formula. We get

$V = (3\text{in})^3$

$V = 3\text{in} \cdot 3\text{in} \cdot 3\text{in}$

$V = 9\text{in}^2 \cdot 3\text{in}$

$V = 27\text{in}^3$

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Example Question #13 : Cubes

Find the area of a cube with a length of 5cm.

Possible Answers:

$25\text{cm}^2$

$15\text{cm}^2$

$150\text{cm}^2$

$\text{There is not enough information to solve the problem.}$

$125\text{cm}^3$

Correct answer:

$125\text{cm}^3$

Explanation:

To find the volume of a cube, we will use the following formula:

$V = l \cdot w \cdot h$

where l is the length, w is the width, and h is the height of the cube.

Now, we know the length of the cube is 5cm. Because it is a cube, all sides are equal. Therefore, the width and the height of the cube are also 5cm.

Knowing this, we will substitute into the formula. We get

$V = 5\text{cm} \cdot5\text{cm} \cdot5\text{cm}$

$V = 25\text{cm}^2 \cdot 5\text{cm}$

$V = 125\text{cm}^3$

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Example Question #14 : Cubes

Find the volume of a cube with a height of 8in.

Possible Answers:

$24\text{in}^3$

$236\text{in}^3$

$325\text{in}^3$

$512\text{in}^3$

$426\text{in}^3$

Correct answer:

$512\text{in}^3$

Explanation:

To find the volume of a cube, we will use the following formula:

$V = l \cdot w \cdot h$

where l is the length, w is the width, and h is the height of the cube.

Now, we know the height of the cube is 8in. Because it is a cube, all sides/lengths are equal. Therefore, the length and width are also 8in.

Knowing this, we can substitute into the formula. We get

$V = 8\text{in} \cdot 8\text{in} \cdot 8\text{in}$

$V = 64\text{in}^2 \cdot 8\text{in}$

$V = 512\text{in}^3$

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Example Question #15 : Cubes

While exploring an ancient ruin, you discover a small puzzle cube. You measure the side length to be . Find the cube's volume.

Possible Answers:

864 cm^3

144 cm^3

1728 cm^3

144 cm^2

Correct answer:

1728 cm^3

Explanation:

While exploring an ancient ruin, you discover a small puzzle cube. You measure the side length to be . Find the cube's volume.

To find the volume of a cube, use the following formula:

V=s^3

V=(12cm)^3=1728cm^3

So our answer is

1728 cm^3

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ISEE Upper Level Math : Solid Geometry

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

All ISEE Upper Level Math Resources

Example Questions

Example Question #1 : Cubes

Example Question #4 : Finding Volume Of A Cube

Example Question #1 : Finding Volume Of A Rectangular Prism

Example Question #1 : How To Find The Volume Of A Cube

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

Example Question #11 : How To Find The Volume Of A Cube

Example Question #12 : Cubes

Example Question #13 : Cubes

Example Question #14 : Cubes

Example Question #15 : Cubes

All ISEE Upper Level Math Resources

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