How to find the volume of a cube - ISEE Upper Level Quantitative Reasoning

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Question

Which is the greater quantity?

(a) The volume of a cube with surface area inches

(b) The volume of a cube with diagonal inches

Answer

The cube with the greater sidelength has the greater volume, so we need only calculate and compare sidelengths.

(a) $A=6s^{2}$ , so the sidelength of the first cube can be found as follows:

$A=6s^{2}$

$6s^{2}= 864$

$6s^{2} \div 6= 864 \div 6$

$s^{2} = 144$

$s = \sqrt{144 }= 12$ inches

(b) $d^{2} = s^{2}+ s^{2}+ s^{2} = 3 s^{2}$ by an extension of the Pythagorean Theorem, so the sidelength of the second cube can be found as follows:

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

$3 s^{2}= 21^{2}= 441$

$3 s^{2}\div 3= 441 \div 3$

$s^{2}= 147$

$s=\sqrt{ 147}$

Since , $\sqrt{147 }> \sqrt{144}$ . The second cube has the greater sidelength and, subsequently, the greater volume. This makes (b) greater.

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Question

Cube 2 has twice the sidelength of Cube 1; Cube 3 has twice the sidelength of Cube 2; Cube 4 has twice the sidelength of Cube 3.

Which is the greater quantity?

(a) The mean of the volumes of Cube 1 and Cube 4

(b) The mean of the volumes of Cube 2 and Cube 3

Answer

The sidelengths of Cubes 1, 2, 3, and 4 can be given values , respectively.

Then the volumes of the cubes are as follows:

Cube 1: $V= s^{3}$

Cube 2: $V= (2s)^{3} = 8s^{3}$

Cube 3: $V= (4s)^{3} = 64s^{3}$

Cube 4: $V= (8s)^{3} = 512s^{3}$

In both answer choices ask for a mean, so we can determine which answer (mean) is greater simply by comparing the sums of volumes.

(a) The sum of the volumes of Cubes 1 and 4 is $s^{3}+ 512^{3} = 513s^{3}$ .

(b) The sum of the volumes of Cubes 2 and 3 is $8s^{3}+ 64^{3} = 72s^{3}$ .

Regardless of , the sum of the volumes of Cubes 1 and 4 is greater, and therefore, so is their mean.

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Question

What is the volume of a cube on which one face has a diagonal of ?

Answer

One of the faces of the cube could be drawn like this:

Notice that this makes a triangle.

This means that we can create a proportion for the sides. On the standard triangle, the non-hypotenuse sides are both , and the hypotenuse is $\sqrt{2}$ . This will allow us to make the proportion:

$\frac{1}{\sqrt{2}} = \frac{x}{2}$

Multiplying both sides by , you get:

$x=\frac{2}{\sqrt{2}}$

Recall that the formula for the volume of a cube is:

Therefore, we can compute the volume using the side found above:

$V = (\frac{2}{\sqrt{2}})^3=\frac{8}{(\sqrt{2})^3}=\frac{8}{2\sqrt{2}}=\frac{4}{\sqrt2}$

Now, rationalize the denominator:

$\frac{4}{\sqrt2} * \frac{\sqrt{2}}{\sqrt{2}} = \frac{4\sqrt{2}}{2}= 2\sqrt{2}$

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Question

What is the volume of a cube with side length ? Round your answer to the nearest hundredth.

Answer

This question is relatively straightforward. The equation for the volume of a cube is:

(It is like doing the area of a square, then adding another dimension!)

Now, for our data, we merely need to "plug and chug:"

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Question

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?

Answer

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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Question

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

Answer

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

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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Question

If a cube is inches tall, what is its volume?

Answer

To find the volume of a cube, we multiply length by width by height, which can be represented with the forumla $v=l\cdot w\cdot h$ . Since a cube has equal sides, we can use for all three values.

$8\cdot 8\cdot 8=512: in ^{3}$

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Question

What is the volume of a cube with a side length equal to inches?

Answer

The volume of a a cube (or rectangular prism) can be solved using the following equation:

$V=l\times\ w\times\ h$

$V=7in\times 7in\times 7in=343in^3$

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Question

What is the volume of a cube in which the edge is equal to $\sqrt[3]{x}$ , and the value of is:

$x=5(4+3^{2})-3$

Answer

First, the value of x must be solved for:

$x=5(4+3^{2})-3$

Given the edge of the cube is $\sqrt[3]{x}$ , plugging in the value of x results in $\sqrt[3]{62}$ . Thus, the area would be equal to this value cubed, which would result in 62.

Thus, 62 is the correct answer.

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Question

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

Answer

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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Question

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

Answer

A cube has a side length of , 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:

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Question

Give the volume of a cube with surface area 150 square inches.

Answer

Let be the length of one edge of the cube. Since its surface area is 150 square inches, one face has one-sixth of this area, or $\frac{150}{6} = 25$ square inches. Therefore, $s^{2} = 25$ , and .

The volume is the cube of this, or $V= 5^{3}=125$ cubic inches.

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Question

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.

Answer

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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Question

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

Answer

Since a diagonal of a square face of the cube is, each side of each square has length $\frac{N}{\sqrt{2}}=\frac{N\sqrt{2}}{2}$ .

Cube this to get the volume of the cube:

$\left (\frac{N\sqrt{2}}{2} \right )^{3}$

$= \frac{N^{3}\left ( \sqrt{2} \right )^{3} }{2^{3} }$

$= \frac{N^{3} \cdot 2 \sqrt{2} }{8}$

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

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Question

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

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

Answer

A perfect cube has square faces; if a face has perimeter meters, then each side of each face measures one fourth of this, or 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.

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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Question

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?

Answer

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:

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Question

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

Answer

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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Question

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

Answer

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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Question

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

Answer

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:

So our answer is

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Question

You are building a box to hold your collection of rare rocks. You want to build a cube-shaped box with a side length of 3 feet. If you do so, what will the volume of your box be?

Answer

You are building a box to hold your collection of rare rocks. You want to build a cube-shaped box with a side length of 3 feet. If you do so, what will the volume of your box be?

Begin with the formula for volume of a cube:

Where s is our side length and V is our volume.

Now, we need to plug in our side length and solve for V

So, our volume is

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