How to find the volume of a cube - Math

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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) , so the sidelength of the first cube can be found as follows:

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

(b) 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$

$s=$\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 .

(b) The sum of the volumes of Cubes 2 and 3 is .

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

The side length of a cube is ft.

What is the volume?

Answer

The volume of a cube is

So with a side length of 2 ft, the volume is

$V = $2^{3}$, $ft^3$ = 8, $ft^3$$

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Question

A cube has edges that are three times as long as those of a smaller cube. The volume of the bigger cube is how many times larger than that of the smaller cube?

Answer

If we let represent the length of an edge on the smaller cube, its volume is .

The larger cube has edges three times as long, so the length can be represented as . The volume is , which is .

The large cube's volume of is 27 times as large as the small cube's volume of .

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Question

The side length of a cube is inches.

What is the volume?

Answer

The volume of a cube is

So with a side length of 7 inches, the volume is

$V = $7^{3}$, $in^3$ = 343, $in^3$$

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Question

Find the volume of a cube with an edge length of .

Answer

The volume of a cube can be determined through the equation , where stands for the length of one side of the cube. The equation is $s\cdot s\cdot s$ because all edges in a cube are the same length. The value for the given edge just needs to be substituted into the equation for in order to solve for the cube's volume.

$V = 7 \cdot 7\cdot 7$

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Question

If the side of a cube is in length, what is the volume of the cube?

Answer

To find the volume of a cube we just multiply a length of one of the cube's sides by itself three times, or in other words, cube it. If we call the length of a side of a cube , then:

The side of our cube is in length, so we can substitute this into the equation and solve:

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Question

A Rubik's Cube is made up of three square layers, with nine identical smaller cubes in each layer. If one of the smaller cubes has side lenghts of 1.5 cm, what is the approximate volume of a whole Rubik's Cube?

Answer

This is a great problem becaue ther are two ways to approach it. The simplest way is to realize that because each edge of the Rubik's Cube is made up of 3 of the smaller cubes we can find the edge length for the whole Rubik's Cube:

(edge of whole Rubik's Cube)

Now that we know the edge length finding the volume is easy, we simply multiply the length, width, and height of the cube to find the volume. This is easy because the length, width, and height are all 4.5 cm.

$4.5 ; cm \times 4.5 ; cm \times 4.5 ; cm = 91.125 ; $cm^3$$ (vol. of Rubik's Cube)

This is the answer. Another way to approach the problem would be by finding the volume of one of the smaller cubes, then multiplying by 9 to find the volume of one layer, then multiplying by three, because there are 3 layers.

$1.5 ; cm \times 1.5 ; cm \times 1.5 ; cm = 3.375; $cm^3$$ (vol. of one small cube)

$3.375 ; $cm^3$ \times 9 = 30.375 $;cm^3$$ (vol. of one layer of Rubik's Cube)

$30.375 ; $cm^3$ \times 3 = 91.125 ; $cm^3$$ (vol. of Rubik's Cube)

Two different methods and we got the same answer!

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Question

A cylinder is cut out of a cube as shown by the figure below.

Find the volume of the figure.

Answer

In order to find the volume of the figure, we must first find the volumes of the cube and the cylinder.

Recall how to find the volume of a cube.

$\text{Volume of $Cube}$=$\text{side}$^3$

Plug in the given side length of the cube to find the volume.

$$\text{Volume of $Cube}$=8^3$=512$

Next, recall how to find the volume of a cylinder.

$\text{Volume of $Cylinder}$=\pi\times$\text{radius}$^2$\times$\text{height}$

Use the given diameter to find the length of the radius.

$$\text{Radius}$=\frac{\text{Diameter}$}{2}=\frac{6}{2}$=3$

Find the volume of the cylinder.

$$\text{Volume of Cylinder}$=\pi\times $3^2$ \times 8=72\pi$

To find the volume of the figure, subtract the volume of the cylinder from the volume of the cube.

$$\text{Volume of Figure}$=512-72\pi=285.81$

Make sure to round to places after the decimal.

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Question

A cylinder is cut out of a cube as shown by the figure below.

Find the volume of the figure.

Answer

In order to find the volume of the figure, we must first find the volumes of the cube and the cylinder.

Recall how to find the volume of a cube.

$\text{Volume of $Cube}$=$\text{side}$^3$

Plug in the given side length of the cube to find the volume.

$$\text{Volume of $Cube}$=10^3$=1000$

Next, recall how to find the volume of a cylinder.

$\text{Volume of $Cylinder}$=\pi\times$\text{radius}$^2$\times$\text{height}$

Use the given diameter to find the length of the radius.

$$\text{Radius}$=\frac{\text{Diameter}$}{2}=\frac{5}{2}$=2.5$

Find the volume of the cylinder.

$$\text{Volume of Cylinder}$=\pi\times $2.5^2$ \times 10=62.5\pi$

To find the volume of the figure, subtract the volume of the cylinder from the volume of the cube.

$$\text{Volume of Figure}$=1000-62.5\pi=803.65$

Make sure to round to places after the decimal.

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Question

A cylinder is cut out of a cube as shown by the figure below.

Find the volume of the figure.

Answer

In order to find the volume of the figure, we must first find the volumes of the cube and the cylinder.

Recall how to find the volume of a cube.

$\text{Volume of $Cube}$=$\text{side}$^3$

Plug in the given side length of the cube to find the volume.

$$\text{Volume of $Cube}$=12^3$=1728$

Next, recall how to find the volume of a cylinder.

$\text{Volume of $Cylinder}$=\pi\times$\text{radius}$^2$\times$\text{height}$

Use the given diameter to find the length of the radius.

$$\text{Radius}$=\frac{\text{Diameter}$}{2}=\frac{8}{2}$=4$

Find the volume of the cylinder.

$$\text{Volume of Cylinder}$=\pi\times $4^2$ \times 12=192\pi$

To find the volume of the figure, subtract the volume of the cylinder from the volume of the cube.

$$\text{Volume of Figure}$=1728-192\pi=1124.81$

Make sure to round to places after the decimal.

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Question

A cylinder is cut out of a cube as shown by the figure below.

Find the volume of the figure.

Answer

In order to find the volume of the figure, we must first find the volumes of the cube and the cylinder.

Recall how to find the volume of a cube.

$\text{Volume of $Cube}$=$\text{side}$^3$

Plug in the given side length of the cube to find the volume.

$$\text{Volume of $Cube}$=4^3$=64$

Next, recall how to find the volume of a cylinder.

$\text{Volume of $Cylinder}$=\pi\times$\text{radius}$^2$\times$\text{height}$

Use the given diameter to find the length of the radius.

$$\text{Radius}$=\frac{\text{Diameter}$}{2}=\frac{1}{2}$=0.5$

Find the volume of the cylinder.

$$\text{Volume of Cylinder}$=\pi\times $0.5^2$ \times 4=\pi$

To find the volume of the figure, subtract the volume of the cylinder from the volume of the cube.

$$\text{Volume of Figure}$=64-\pi=60.86$

Make sure to round to places after the decimal.

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Question

A cylinder is cut out of a cube as shown by the figure below.

Find the volume of the figure.

Answer

In order to find the volume of the figure, we must first find the volumes of the cube and the cylinder.

Recall how to find the volume of a cube.

$\text{Volume of $Cube}$=$\text{side}$^3$

Plug in the given side length of the cube to find the volume.

$$\text{Volume of $Cube}$=5^3$=125$

Next, recall how to find the volume of a cylinder.

$\text{Volume of $Cylinder}$=\pi\times$\text{radius}$^2$\times$\text{height}$

Use the given diameter to find the length of the radius.

$$\text{Radius}$=\frac{\text{Diameter}$}{2}=\frac{3}{2}$=1.5$

Find the volume of the cylinder.

$$\text{Volume of Cylinder}$=\pi\times $1.5^2$ \times 5=11.25\pi$

To find the volume of the figure, subtract the volume of the cylinder from the volume of the cube.

$$\text{Volume of Figure}$=125-11.25\pi=89.66$

Make sure to round to places after the decimal.

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Question

A cylinder is cut out of a cube as shown by the figure below.

Find the volume of the figure.

Answer

In order to find the volume of the figure, we must first find the volumes of the cube and the cylinder.

Recall how to find the volume of a cube.

$\text{Volume of $Cube}$=$\text{side}$^3$

Plug in the given side length of the cube to find the volume.

$$\text{Volume of $Cube}$=9^3$=729$

Next, recall how to find the volume of a cylinder.

$\text{Volume of $Cylinder}$=\pi\times$\text{radius}$^2$\times$\text{height}$

Use the given diameter to find the length of the radius.

$$\text{Radius}$=\frac{\text{Diameter}$}{2}=\frac{8}{2}$=4$

Find the volume of the cylinder.

$$\text{Volume of Cylinder}$=\pi\times $4^2$ \times 9=144\pi$

To find the volume of the figure, subtract the volume of the cylinder from the volume of the cube.

$$\text{Volume of Figure}$=729-144\pi=276.61$

Make sure to round to places after the decimal.

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Question

A cylinder is cut out of a cube as shown by the figure below.

Find the volume of the figure.

Answer

In order to find the volume of the figure, we must first find the volumes of the cube and the cylinder.

Recall how to find the volume of a cube.

$\text{Volume of $Cube}$=$\text{side}$^3$

Plug in the given side length of the cube to find the volume.

$$\text{Volume of $Cube}$=15^3$=3375$

Next, recall how to find the volume of a cylinder.

$\text{Volume of $Cylinder}$=\pi\times$\text{radius}$^2$\times$\text{height}$

Use the given diameter to find the length of the radius.

$$\text{Radius}$=\frac{\text{Diameter}$}{2}=\frac{10}{2}$=5$

Find the volume of the cylinder.

$$\text{Volume of Cylinder}$=\pi\times $5^2$ \times 15=375\pi$

To find the volume of the figure, subtract the volume of the cylinder from the volume of the cube.

$$\text{Volume of Figure}$=3375-375\pi=2196.90$

Make sure to round to places after the decimal.

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Question

A cylinder is cut out of a cube as shown by the figure below.

Find the volume of the figure.

Answer

In order to find the volume of the figure, we must first find the volumes of the cube and the cylinder.

Recall how to find the volume of a cube.

$\text{Volume of $Cube}$=$\text{side}$^3$

Plug in the given side length of the cube to find the volume.

$$\text{Volume of $Cube}$=10^3$=1000$

Next, recall how to find the volume of a cylinder.

$\text{Volume of $Cylinder}$=\pi\times$\text{radius}$^2$\times$\text{height}$

Use the given diameter to find the length of the radius.

$$\text{Radius}$=\frac{\text{Diameter}$}{2}=\frac{8}{2}$=4$

Find the volume of the cylinder.

$$\text{Volume of Cylinder}$=\pi\times $4^2$ \times 10=160\pi$

To find the volume of the figure, subtract the volume of the cylinder from the volume of the cube.

$$\text{Volume of Figure}$=1000-160\pi=497.35$

Make sure to round to places after the decimal.

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Question

A cylinder is cut out of a cube as shown by the figure below.

Find the volume of the figure.

Answer

In order to find the volume of the figure, we must first find the volumes of the cube and the cylinder.

Recall how to find the volume of a cube.

$\text{Volume of $Cube}$=$\text{side}$^3$

Plug in the given side length of the cube to find the volume.

$$\text{Volume of $Cube}$=15^3$=3375$

Next, recall how to find the volume of a cylinder.

$\text{Volume of $Cylinder}$=\pi\times$\text{radius}$^2$\times$\text{height}$

Use the given diameter to find the length of the radius.

$$\text{Radius}$=\frac{\text{Diameter}$}{2}=\frac{8}{2}$=4$

Find the volume of the cylinder.

$$\text{Volume of Cylinder}$=\pi\times $4^2$ \times 15=240\pi$

To find the volume of the figure, subtract the volume of the cylinder from the volume of the cube.

$$\text{Volume of Figure}$=3375-240\pi=2621.02$

Make sure to round to places after the decimal.

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Question

A cylinder is cut out of a cube as shown by the figure below.

Find the volume of the figure.

Answer

In order to find the volume of the figure, we must first find the volumes of the cube and the cylinder.

Recall how to find the volume of a cube.

$\text{Volume of $Cube}$=$\text{side}$^3$

Plug in the given side length of the cube to find the volume.

$$\text{Volume of $Cube}$=16^3$=4096$

Next, recall how to find the volume of a cylinder.

$\text{Volume of $Cylinder}$=\pi\times$\text{radius}$^2$\times$\text{height}$

Use the given diameter to find the length of the radius.

$$\text{Radius}$=\frac{\text{Diameter}$}{2}=\frac{8}{2}$=4$

Find the volume of the cylinder.

$$\text{Volume of Cylinder}$=\pi\times $4^2$ \times 16=256\pi$

To find the volume of the figure, subtract the volume of the cylinder from the volume of the cube.

$$\text{Volume of Figure}$=4096-256\pi=3291.75$

Make sure to round to places after the decimal.

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