SAT Math : Geometry

Study concepts, example questions & explanations for SAT Math

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Example Questions

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

A room has dimensions of 23ft by 17ft by 10ft. The last dimension is the height of the room. It has one door that is 2.5ft by 8ft and one window, 3ft by 6ft. There is no trim to the floor, wall, doors, or windows. If one can of paint covers 57 ft2 of surface area. How many cans of paint must be bought to paint the walls of the room.

Possible Answers:

11

15

13

14

18

Correct answer:

14

Explanation:

If broken down into parts, this is an easy problem. It is first necessary to isolate the dimensions of the walls. If the room is 10ft high, we know 23 x 17 designates the area of the floor and ceiling. Based on this, we know that the room has the following dimensions for the walls: 23 x 10 and 17 x 10. Since there are two of each, we can calculate the total area of walls - ignoring doors and windows - by doubling the sum of these two areas:

2 * (23 * 10 + 17 * 10) = 2 * (230 + 170) = 2 * 400 = 800 ft2

Now, we merely need to calculate the area "taken out" of the walls:

For the door: 2.5 * 8 = 20 ft2 

For the windows: 3 * 6 = 18 ft2

The total wall space is therefore: 800 – 20 – 18 = 762 ft2

Now, if one can of paint covers 57 ft2, we calculate the number of cans necessary by dividing the total exposed area by 57: 762/57 = (approx.) 13.37.

Since we cannot buy partial cans, we must purchase 14 cans.

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

A cubic box has sides of length x. Another cubic box has sides of length 4x. How many of the boxes with length x could fit in one of the larger boxes with side length 4x?

Possible Answers:

80

16

40

4

64

Correct answer:

64

Explanation:

The volume of a cubic box is given by (side length)3. Thus, the volume of the larger box is (4x)3 = 64x3, while the volume of the smaller box is x3. Divide the volume of the larger box by that of the smaller box, (64x3)/(x3) = 64.

Example Question #2 : Cubes

I have a hollow cube with 3” sides suspended inside a larger cube of 9” sides.  If I fill the larger cube with water and the hollow cube remains empty yet suspended inside, what volume of water was used to fill the larger cube?

Possible Answers:

72 in3

702 in3

73 in3

216 in3

698 in3

Correct answer:

702 in3

Explanation:

Determine the volume of both cubes and then subtract the smaller from the larger.  The large cube volume is 9” * 9” * 9” = 729 in3 and the small cube is 3” * 3” * 3” = 27 in3.  The difference is 702 in3.

Example Question #3 : Cubes

A cube weighs 5 pounds. How much will a different cube of the same material weigh if the sides are 3 times as long?

Possible Answers:

15 pounds

45 pounds

135 pounds

10 pounds

Correct answer:

135 pounds

Explanation:

A cube that has three times as long sides is 3x3x3=27 times bigger than the original. Therefore, the answer is 5x27= 135.

Example Question #1 : Cubes

If the volume of a cube is 50 cubic feet, what is the volume when the sides double in length?

Possible Answers:

500 cu ft

300 cu ft

100 cu ft

400 cu ft

200 cu ft

Correct answer:

400 cu ft

Explanation:

Using S as the side length in the original cube, the original is s*s*s. Doubling one side and tripling the other gives 2s*2s*2s for a new volume formula for 8s*s*s, showing that the new volume is 8x greater than the original.

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

A cube has 2 faces painted red and the remaining faces painted green. The total area of the green faces is 36 square inches. What is the volume of the cube in cubic inches?

Possible Answers:

8

64

16

27

54

Correct answer:

27

Explanation:

Cubes have 6 faces. If 2 are red, then 4 must be green. We are told that the total area of the green faces is 36 square inches, so we divide the total area of the green faces by the number of green faces (which is 4) to get the area of each green face: 36/4 = 9 square inches. Since each of the 6 faces of a cube have the same size, we know that each edge of the cube is √9 = 3 inches. Therefore the volume of the cube is 3 in x 3 in x 3 in = 27 cubic inches.

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

If a waterproof box is 50cm in length, 20cm in depth, and 30cm in height, how much water will overflow if 35 liters of water are poured into the box?

Possible Answers:

15 liters

1 liters

30 lites

5 liters

No water will flow out of the box

Correct answer:

5 liters

Explanation:

The volume of the box is 50 * 20 * 30 cm = 30,000 cm3.

1cm3 = 1mL, 30,000 cm3 = 30,000mL = 30 L.

Because the volume of the box is only 30 L, 5 L of the 35 L will not fit into the box.

Example Question #31 : Cubes

Kim from Idaho can only stack bales of hay in her barn for 3 hours before she needs a break. She stacks the bales at a rate of 2 bales per minute, 3 bales high with 5 bales in a single row. How many full rows will she have at the end of her stacking?

Possible Answers:

15

27

24

20

16

Correct answer:

24

Explanation:

She will stack 360 bales in 3 hours. One row requires 15 bales. 360 divided by 15 is 24. 

Example Question #31 : Solid Geometry

A cube has a volume of \dpi{100} \small 8 cm^{3}. What is the volume of cube with sides that are twice as long?

Possible Answers:

\dpi{100} \small 2 cm^{3}

\dpi{100} \small 12 cm^{3}

\dpi{100} \small 27 cm^{3}

\dpi{100} \small 64 cm^{3}

\dpi{100} \small 16 cm^{3}

Correct answer:

\dpi{100} \small 64 cm^{3}

Explanation:

The volume of a cube is \dpi{100} \small s^{3}.

If each side of the cube is \dpi{100} \small 2cm, then the volume will be \dpi{100} \small 8cm^{3}.

If we double each side, then each side would be \dpi{100} \small 4cm and the volume would be \dpi{100} \small 64cm^{3}.

Example Question #1381 : Concepts

How many  smaller boxes with a dimensions of  1 by 5 by 5 can fit into cube shaped box with a surface area of 150?

Possible Answers:

7

5

6

8

4

Correct answer:

5

Explanation:

There surface are of a cube is 6 times the area of one face of the cube , therefore 6a^{2}=150

a^{2}=25

a=5

a is equal to an edge of the cube

the volume of the cube is a^{3}=5^{3}=125

The problem states that the dimensions of the smaller boxes are 1 x 5 x 5, the volume of one of the smaller boxes is 25.

Therefore, 125/25 = 5 small boxes

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