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ACT Math : Geometry

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Example Questions

Example Question #582 : Geometry

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

Possible Answers:

$54\ cm^2$

$25\ cm^2$

$45\ cm^2$

$63\ cm^2$

$36\ cm^2$

Correct answer:

$54\ cm^2$

Explanation:

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

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

Act4

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

Possible Answers:

$48\ m^{2}$

$96\ m^{2}$

$24\ m^{2}$

$48\pi \ m^{2}$

$48\pi ^{2}\ m^{2}$

Correct answer:

$96\ m^{2}$

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.

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

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

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

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

$8=r^{3}$

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

$side=2\cdot 2=4$

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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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:

$64\:in^2$

$24\:in^2$

$96\:in^2$

$384\:in^2$

$192\:in^2$

Correct answer:

$96\:in^2$

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 , 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 $96\:in^2$ .

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

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

Possible Answers:

256mm^2

135.76mm^2

64mm^2

384mm^2

90.51mm^2

Correct answer:

384mm^2

Explanation:

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

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

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

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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 4 inches ?

Possible Answers:

64in^2

32in^2

16in^2

96in^2

12in^2

Correct answer:

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

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

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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:

s = 10

s = 6

There is not enough information to determine the answer.

$s = \sqrt{6}$

s = 4

Correct answer:

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:

V=s^3: SA=6a^2

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

$\\ s^3 = 6*s^2 \\ \newline \frac{s^3}{s^2}=\frac{6*s^2}{s^2}\\ \newline s = 6$

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

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

Possible Answers:

$9\textup{ ft}^2$

$162\textup{ ft}^2$

$27\textup{ ft}^2$

$54\textup{ ft}^2$

$18\textup{ ft}^2$

Correct answer:

$54\textup{ ft}^2$

Explanation:

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

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

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

Possible Answers:

Correct answer:

Explanation:

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

SA=s^2*6=4^2*6=16*6=96

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Example Question #21 : Cubes

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

Possible Answers:

108

Correct answer:

Explanation:

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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Example Question #22 : Cubes

Find the surface area of a cube with side length .

Possible Answers:

Correct answer:

Explanation:

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

SA=s^2*6=1^2*6=6

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ACT Math : Geometry

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #582 : Geometry

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

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

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

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

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

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

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

Example Question #21 : Cubes

Example Question #22 : Cubes

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