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

Study concepts, example questions & explanations for ACT Math

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Example Questions

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

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

Possible Answers:

5

3

6

4

Correct answer:

4

Explanation:

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

Solving we get √16, so a = 4

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

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

Possible Answers:

\dpi{100} \small 5cm^{2}\(\displaystyle \dpi{100} \small 5cm^{2}\)

\dpi{100} \small 50 cm^{2}\(\displaystyle \dpi{100} \small 50 cm^{2}\)

\dpi{100} \small 150cm^{2}\(\displaystyle \dpi{100} \small 150cm^{2}\)

\dpi{100} \small 25 cm^{2}\(\displaystyle \dpi{100} \small 25 cm^{2}\)

\dpi{100} \small 125 cm^{2}\(\displaystyle \dpi{100} \small 125 cm^{2}\)

Correct answer:

\dpi{100} \small 150cm^{2}\(\displaystyle \dpi{100} \small 150cm^{2}\)

Explanation:

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\(\displaystyle \dpi{100} \small 5\times 5 = 25\)

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\(\displaystyle \dpi{100} \small 25 \times 6=150\)

Example Question #2 : How To Find The Surface Area Of A Cube

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

Possible Answers:

\(\displaystyle 45\ cm^2\)

\(\displaystyle 25\ cm^2\)

\(\displaystyle 36\ cm^2\)

\(\displaystyle 54\ cm^2\)

\(\displaystyle 63\ cm^2\)

Correct answer:

\(\displaystyle 54\ cm^2\)

Explanation:

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

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

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

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

Example Question #3 : How To Find The Surface Area Of A Cube

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

Act4

What is the surface area of the cube, in \(\displaystyle m^{2}\)?

Possible Answers:

\(\displaystyle 48\ m^{2}\)

\(\displaystyle 48\pi \ m^{2}\)

\(\displaystyle 96\ m^{2}\)

\(\displaystyle 24\ m^{2}\)

\(\displaystyle 48\pi ^{2}\ m^{2}\)

Correct answer:

\(\displaystyle 96\ m^{2}\)

Explanation:

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.

\(\displaystyle V_{sphere}=\frac{4}{3}\pi r^{3}\)

\(\displaystyle \frac{32}{3}\pi =\frac{4}{3}\pi r^{3}\)

\(\displaystyle \frac{32}{3}=\frac{4}{3}r^{3}\)

\(\displaystyle \frac{3}{4}\cdot \frac{32}{3}= r^{3}\)

\(\displaystyle 8=r^{3}\)

\(\displaystyle r=2\)

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

\(\displaystyle side=2\cdot 2=4\)

The formula for the surface area of a cube is:

\(\displaystyle SA_{cube}=6s^{2}\)

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

The surface area of the cube is \(\displaystyle 96\ m^{2}\)

 

Example Question #3 : How To Find The Surface Area Of A Cube

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

Possible Answers:

\(\displaystyle 24\:in^2\)

\(\displaystyle 96\:in^2\)

\(\displaystyle 384\:in^2\)

\(\displaystyle 192\:in^2\)

\(\displaystyle 64\:in^2\)

Correct answer:

\(\displaystyle 96\:in^2\)

Explanation:

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 \(\displaystyle 6\), because a cube has \(\displaystyle 6\) 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:

\(\displaystyle area = L\cdot W=4\:in\cdot 4\:in=16\:in^2\)

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

\(\displaystyle 16\:in^2\cdot6=96\:in^2\)

Therefore, the surface are of a four-inch cube is \(\displaystyle 96\:in^2\).

Example Question #4 : How To Find The Surface Area Of A Cube

What is the surface area of a cube with a volume of \(\displaystyle 512mm^3\)? Round your answer to the nearest hundreth if necessary

Possible Answers:

\(\displaystyle 256mm^2\)

\(\displaystyle 135.76mm^2\)

\(\displaystyle 64mm^2\)

\(\displaystyle 384mm^2\)

\(\displaystyle 90.51mm^2\)

Correct answer:

\(\displaystyle 384mm^2\)

Explanation:

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

\(\displaystyle \sqrt[n]{512mm^3}\) = \(\displaystyle 8mm\)
Next plug the side length into the formula for the surface area of a cube:

\(\displaystyle SA = 6*s^2 = 6 *(8mm)^2 = 384mm^2\)

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

What is the surface area, in square inches, of a cube with sides measuring \(\displaystyle 4 inches\) ?

Possible Answers:

\(\displaystyle 64in^2\)

\(\displaystyle 32in^2\)

\(\displaystyle 16in^2\)

\(\displaystyle 96in^2\)

\(\displaystyle 12in^2\)

Correct answer:

\(\displaystyle 96in^2\)

Explanation:

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

 

Since a cube contains \(\displaystyle 6\) square sides, the surface area is \(\displaystyle 6\) times the area of a square side.

 

The area of one square side is sidelength \(\displaystyle x\) sidelength, or \(\displaystyle 4*4 = 16\) in this case. Therefore, the surface area of this cube is \(\displaystyle 16*6 = 96\) square inches.

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

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

Possible Answers:

\(\displaystyle s = 4\)

\(\displaystyle s = 10\)

There is not enough information to determine the answer.

\(\displaystyle s = 6\)

\(\displaystyle s = \sqrt{6}\)

Correct answer:

\(\displaystyle s = 6\)

Explanation:

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:

\(\displaystyle V=s^3: SA=6a^2\)

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

\(\displaystyle \\ s^3 = 6*s^2 \\ \newline \frac{s^3}{s^2}=\frac{6*s^2}{s^2}\\ \newline s = 6\)

Example Question #6 : How To Find The Surface Area Of A Cube

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

Possible Answers:

\(\displaystyle 27\textup{ ft}^2\)

\(\displaystyle 54\textup{ ft}^2\)

\(\displaystyle 9\textup{ ft}^2\)

\(\displaystyle 162\textup{ ft}^2\)

\(\displaystyle 18\textup{ ft}^2\)

Correct answer:

\(\displaystyle 54\textup{ ft}^2\)

Explanation:

To find the surface area of a cube with a given side length, \(\displaystyle s\) use the formula:
\(\displaystyle SA = 6s^2 = 6*3^2 = 6*9 = 54\)

Example Question #4 : How To Find The Surface Area Of A Cube

Find the surface area of a cube whose side length is \(\displaystyle 4\).

Possible Answers:

\(\displaystyle 96\)

\(\displaystyle 48\)

\(\displaystyle 64\)

\(\displaystyle 24\)

\(\displaystyle 16\)

Correct answer:

\(\displaystyle 96\)

Explanation:

To find surface area of a cube, simply calculate the area of one side and multiply it by \(\displaystyle 6\). Thus,

 \(\displaystyle SA=s^2*6=4^2*6=16*6=96\)

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