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

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

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

$3N^{3}$

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

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

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

Explanation

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:

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

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

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

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

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.

The correct answer is not given among the other responses.

$375 \textup{ L}$

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

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

$750\textup{ L}$

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.

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

Explanation

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:

Find the volume of a cube with a length of 5in.

$125\text{in}^3$

$75\text{in}^3$

$100\text{in}^3$

$150\text{in}^3$

$175\text{in}^3$

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 length of the cube is 5in. Because it is a cube, all sides/lengths are equal. Therefore, the length and width are also 5in.

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

$V = 5\text{in} \cdot 5\text{in} \cdot 5\text{in}$

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

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

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

$512\text{in}^3$

$24\text{in}^3$

$426\text{in}^3$

$325\text{in}^3$

$236\text{in}^3$

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

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

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

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

Find the volume of a cube with a height of 7in.

$343\text{in}^3$

$147\text{in}^3$

$392\text{in}^3$

$349\text{in}^3$

$221\text{in}^3$

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 7in. Because it is a cube, all sides are equal. Therefore, all sides (the length, width, height) are all 7in.

So, we get

$V = 7\text{in} \cdot 7\text{in} \cdot 7\text{in}$

$V = 49\text{in}^2 \cdot 7\text{in}$

$V = 343\text{in}^3$

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

$125\text{cm}^3$

$15\text{cm}^2$

$25\text{cm}^2$

$150\text{cm}^2$

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

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

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

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

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

Find the volume of a cube with a width of 9in.

$729\text{in}^3$

$648\text{in}^3$

$549\text{in}^3$

$819\text{in}^3$

$436\text{in}^3$

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 width is 9in. Because it is a cube, all sides are equal. Therefore, the length and the height are also 9in.

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

$V = 9\text{in} \cdot9\text{in} \cdot9\text{in}$

$V = 81\text{in}^2 \cdot 9\text{in}$

$V = 729\text{in}^3$

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?

Explanation

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

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

$27\text{in}^3$

$9\text{in}^3$

$18\text{in}^3$

$12\text{in}^3$

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

Explanation

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

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