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SAT Math : Solid Geometry

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Example Question #14 : How To Find The Volume Of A Cube

The volumes of six cubes form an arithmetic sequence. The two smallest cubes have sidelengths 10 and 12, respectively. Give the volume of the largest cube.

Possible Answers:

2,400

8,000

1,920

4,640

5,368

Correct answer:

4,640

Explanation:

The volume of a cube is equal to the length of a side raised to the third power. The two smallest cubes will have volumes:

$V_{1} = 10^{3} = 1,000$

and

$V_{2} = 12^{3} = 1,728$ ,

respectively.

The volumes form an arithmetic sequence with these two volumes as the first two terms, so their common difference is

$d = V_{2} - V_{1} = 1,728 - 1,000 = 728$ .

The volume of the largest, or sixth-smallest, cube, is

$V_{6} = (6-1)d + V_{1}$

=5 (728) + 1,000

=3,640 + 1,000

= 4,640

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Example Question #41 : Solid Geometry

Find the volume of a cube with side length 4.

Possible Answers:

254

Correct answer:

Explanation:

To solve, simply use the formula for the volume of a cube.

Substitute in the side length of four into the following equation.

Thus,

V=s^3=4^3=64

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Example Question #731 : Geometry

Find the volume of a cube given side length is 1.

Possible Answers:

Correct answer:

Explanation:

To solve, simply use the formula for the volume of a cube. Thus,

V=s^3=1^3=1

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Example Question #41 : Cubes

A cube has a surface area of 150 . What is its volume?

Possible Answers:

150

100

125

Correct answer:

125

Explanation:

Remember that a cube's surface area, because it's comprised of six identical squares, can be stated as 6s^2 . With that in mind,

6s^2 = 150

s^2 = 25

$s = \sqrt{25} = 5$

The last step is easy: V = s^3 = 5^3 = 125

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Example Question #11 : How To Find The Volume Of A Cube

What is the volume of a cube with a side length of 7.5 cm?

(Round two the nearest two places)

Possible Answers:

274.63cm^2

512cm^2

857.38cm^3

421.88cm^3

Correct answer:

421.88cm^3

Explanation:

The formula for volume of a cube is,

V=s^3

where

s=side .

The side length of the cube is given as 7.5cm.

s=7.5

Substituting this into the formula for a cube's volume is as follows.

$\\V=(7.5)^3 \\V=(7.5)(7.5)(7.5) \\V=421.875\approx 421.88cm^3$

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Example Question #42 : Cubes

Find the volume of a cube whose side length is 7cm.

Possible Answers:

21cm^3

343cm^3

196cm^3

49cm^3

7cm^3

Correct answer:

343cm^3

Explanation:

The volume of a cube is length*width*height. In a cube all the side lengths are equal. Volume=7cm*7cm*7cm=343cm^3

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Example Question #41 : Cubes

One side of a cube is $\frac{\pi}{2}$ units long. What is the volume of the cube if it is cut in half?

Possible Answers:

$\frac{\pi^2}{16}$

$\frac{\pi}{8}$

$\frac{\pi^3}{16}$

Correct answer:

$\frac{\pi^3}{16}$

Explanation:

The volume of a cube is given by multiplying its length, width, and height, which are all equal. Therefore we can say:

V=lwh = h^3

Substitute in our given measurement:

$V = (\frac{\pi}{2})^3 = \frac{\pi^3}{8}$

We want to know the volume of the cube if it is cut in half, so let's divide this answer in two:

$\frac{\pi^3}{8}\cdot \frac{1}{2} = \frac{\pi^3}{16}$

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Example Question #41 : Solid Geometry

At your university there is a metal cube-shaped sculpture near the math building. If the cube has a side length of 4 meters, what is the cube's volume?

Possible Answers:

16m^2

16m^3

64m^3

64m^2

Correct answer:

64m^3

Explanation:

At your university there is a metal cube-shaped sculpture near the math building. If the cube has a side length of 4 meters, what is the cube's volume?

Begin with the formula for volume of a cube, then just simplify:

$V_{cube}=s^3=(4m)^3=64m^3$

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Example Question #1 : How To Find The Length Of An Edge Of A Pyramid

If the height of a pyramid was increased by 20% and a side of the square base was decreased 30%, what would happen to the volume of the pyramid?

Possible Answers:

It would have the same volume

There is no way to know if it would increase or decrease in volume

There is a 41% decrease in volume

59% decrease in volume

There is a 59% increase in volume

Correct answer:

There is a 41% decrease in volume

Explanation:

First, you will want to create a pyramid with measurements that are easy to calculate. So, let's say that we have pyramid with a base edge of 10 inches and a height of 10 inches.

So the volume of the original pyramid would be equal to

$\frac{(10\times 10)(10)}{3}=\frac{1000}{3}=333.33\: inches^2$

The volume of the altered pyramid would be equal to:

$\frac{(7\times 7)(12)}{3}=\frac{588}{3}=196\: inches^2$

To find the relationship between the volume of the altered pyramid relative to the volume of the original pyramid, divide the altered volume by the original volume.

$\frac{196}{333.33}=.59$

The new volume is 59% of the original volume, which means there was a 41% decrease in volume.

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

The Pyramid of Giza has a height of 480 feet. If the length of each side of the base is approximately 756 feet, what is its total surface area? Round to the nearest tenth.

Possible Answers:

$802,494 ft^{2}$

$752,976 ft^{2}$

$1,495,368 ft^{2}$

$934,416 ft^{2}$

Correct answer:

$1,495,368 ft^{2}$

Explanation:

If the length of one side is 756 ft, then multiply to find the area of the base.

$\text{Area of square} = \text{length} \times \text{width}$

$A=756\times756$

$A=571,536 ft^{2}$

Once you've found the area of the base, use the height of the pyramid and half of the side length of the base to determine the length of the side from the apex to the ground using the Pythagorean Theorem.

$480^{2}+378^{2}=c^{2}$

$230,400+142,884=c^{2}$

$373,284=c^{2}$

611=c

Using the side length of the base and the height of each of the triangles that form the pyramid, calculate the area of each triangle, then multiply by 4.

$\text{Area of a triangle}=\frac{1}{2}\times \text{base} \times \text{height}$

$A=\frac{1}{2} \times 611 \times 756$

A=230,958

$\text{Area of four triangles} = 4 \times 230,958$

$A=923,832 ft^{2}$

Add the surface area of the base to the surface area of the four triangles.

$\text{Total surface area} = 571,536 + 923,832$

$1,495,368 ft^{2}$

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SAT Math : Solid Geometry

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #14 : How To Find The Volume Of A Cube

Example Question #41 : Solid Geometry

Example Question #731 : Geometry

Example Question #41 : Cubes

Example Question #11 : How To Find The Volume Of A Cube

Example Question #42 : Cubes

Example Question #41 : Cubes

Example Question #41 : Solid Geometry

Example Question #1 : How To Find The Length Of An Edge Of A Pyramid

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

All SAT Math Resources

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