How to find the surface area of a cube - ACT Math

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Question

If the surface area of a cube equals 96, what is the length of one side of the cube?

Answer

The surface area of a cube = 6a2 where a is the length of the side of each edge of the cube. Put another way, since all sides of a cube are equal, a is just the lenght of one side of a cube.

We have 96 = 6a2 → a2 = 16, so that's the area of one face of the cube.

Solving we get √16, so a = 4

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Question

A sphere with a volume of $\frac{32}{3}$ $\pi m^{3}$ is inscribed in a cube, as shown in the diagram below.

What is the surface area of the cube, in $m^{2}$ ?

Answer

We must first find the radius of the sphere in order to solve this problem. Since we already know the volume, we will use the volume formula to do this.

$V_{sphere}=\frac{4}{3}\pi r^{3}$

$\frac{32}{3}\pi =\frac{4}{3}\pi r^{3}$

$\frac{32}{3}=\frac{4}{3}r^{3}$

$\frac{3}{4}\cdot \frac{32}{3}= r^{3}$

$8=r^{3}$

With the radius of the sphere in hand, we can now apply it to the cube. The radius of the sphere is half the distance from the top to the bottom of the cube (or half the distance from one side to another). Therefore, the radius represents half of a side length of a square. So in this case

$side=2\cdot 2=4$

The formula for the surface area of a cube is:

$SA_{cube}=6s^{2}$

$6s^{2}=6\cdot (4)^{2}=6\cdot 16=96$

The surface area of the cube is $96\ m^{2}$

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Question

What is the surface area of a cube if its height is 3 cm?

Answer

The area of one face is given by the length of a side squared.

$A_{face}=(3cm)^2=9\ cm^2$

The area of 6 faces is then given by six times the area of one face: 54 cm2.

$A_{total}=6(A_{face})=6(9\ cm^2)=54\ cm^2$

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Question

The side of a cube has a length of $\dpi{100} \small 5 cm$ . What is the total surface area of the cube?

Answer

A cube has 6 faces. The area of each face is found by squaring the length of the side.

$\dpi{100} \small 5\times 5 = 25$

Multiply the area of one face by the number of faces to get the total surface area of the cube.

$\dpi{100} \small 25 \times 6=150$

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Question

What is the surface area, in square inches, of a four-inch cube?

Answer

To answer this question, we need to find the surface area of a cube.

To do this, we must find the area of one face and multiply it by , because a cube has faces that are square in shape and equal in size.

To find the area of a square, you multiply its length by its width. (Note that the length and width of a square are the same.) Therefore, for this data:

$area = L\cdot W=4:in\cdot 4:in=16:in^2$

We now must multiply the area of one face by 6 to get the total surface are of the cube.

$16:in^2\cdot6=96:in^2$

Therefore, the surface are of a four-inch cube is .

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Question

What is the surface area, in square inches, of a cube with sides measuring ?

Answer

The surface area of a cube is a measure of the total area of thesurface of all of the sides of that cube.

Since a cube contains square sides, the surface area is times the area of a square side.

The area of one square side is sidelength sidelength, or in this case. Therefore, the surface area of this cube is square inches.

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Question

What is the surface area of a cube with a volume of ? Round your answer to the nearest hundreth if necessary

Answer

First we need to find the side length of the cube. Do that by taking the cube root of the volume.

$\sqrt[n]{512mm^3}$ =
Next plug the side length into the formula for the surface area of a cube:

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Question

What is the length of the side of a cube whose surface area is equal to its volume?

Answer

To find the side length of a cube whose surface area is the same as its volume, set the surface area and volume equations of a cube equal to each other, the solve for the side length:

Set these two formulas equal to eachother and solve for s.

$\ s^3 = 6s^2 \ ewline \frac{s^3}{s^2}=\frac{6s^2}{s^2}\ ewline s = 6$

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Question

What is the surface area of a cube with a side of length $\textup{3ft}$ ?

Answer

To find the surface area of a cube with a given side length, use the formula:

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Question

Find the surface area of a cube whose side length is .

Answer

To find surface area of a cube, simply calculate the area of one side and multiply it by . Thus,

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Question

Find the surface area of a cube whose side length is .

Answer

To find surface aarea, simply multiply the area of a face by 6 since there are 6 faces. Thus,

$SA=6\cdot s^2=6\cdot3^2=6\cdot9=54$

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Question

Find the surface area of a cube with side length .

Answer

To solve, simply multiply the face area by . Thus,

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