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PSAT Math : Geometry

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Example Questions

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:

Correct answer:

Explanation:

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

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

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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:

Correct answer:

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

If a cube has its edges increased by a factor of 5, what is the ratio of the new volume to the old volume?

Possible Answers:

100:1

15:1

5:1

125:1

25:1

Correct answer:

125:1

Explanation:

A cubic volume is side*side*side . Let the original sides be 1, so that the original volume is 1. Then find the volume if the sides measure 5. This new volume is 125. Therefore, the ratio of new volume to old volume is 125: 1.

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Example Question #221 : Geometry

A cube is inscribed inside a sphere of radius 1 such that each of the eight vertices of the cube lie on the surface of the sphere. What is the volume of the cube?

Possible Answers:

$\frac{7\sqrt{3}}{10}$

$\frac{8\sqrt{3}}{9}$

$\frac{9\sqrt{3}}{8}$

$\frac{10\sqrt{3}}{7}$

Correct answer:

$\frac{8\sqrt{3}}{9}$

Explanation:

Cube

To make this problem easier to solve, we can inscribe a smaller square in the cube. In the diagram above, points A, B, C are midpoints of the edges of the inscribed cube. Therefore point , a vertex of the smaller cube, is also the center of the sphere. Point lies on the circumference of the sphere, so ED = 1 . is also the hypotenuse of right triangle FED . Similarly, is the hypotenuse of right triangle FBD . If we let BD = x , then, by the properties of a right triangle, $FD = x\sqrt2$ .

Using the Pythagorean Theorem, we can now solve for :

FD^2 + FE^2 = DE^2

$(x\sqrt{2})^2 + x^2 = 1^2$

2x^2 + x^2 = 1

3x^2 = 1

$x^2 = \frac{1}{3}$

$x=\sqrt{\frac{1}{3}}$

Since the side of the inscribed cube is , the volume is $V= (2x)^3=8x^3=8\sqrt{\frac{1}{3}}^3=\frac{8}{3\sqrt{3}}=\frac{8}{3\sqrt{3}}\times \frac{\sqrt{3}}{\sqrt{3}}=\frac{8\sqrt{3}}{9}$ .

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Example Question #221 : Psat Mathematics

If a cube has a surface area of 600 , what is the difference between the volume of the cube and the surface area of the cube?

Possible Answers:

200

400

500

600

1000

Correct answer:

400

Explanation:

If the surface area is $600\textup{ sq in}$ , then the area of one face must be $100\textup{ sq in}$ . Therefore, the length of one edge must be $10\textup{ in}$ This means that the volume of the cube is $1000\textup{ cubic in}$ . We can now solve with:

1000-600=400

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Example Question #3 : Solid Geometry

What is the surface area of a cube with a volume of 1728 in³?

Possible Answers:

432 in²

72 in²

144 in²

864 in²

1728 in²

Correct answer:

864 in²

Explanation:

This problem is relatively simple. We know that the volume of a cube is equal to s³, where s is the length of a given side of the cube. Therefore, to find our dimensions, we merely have to solve s³ = 1728. Taking the cubed root, we get s = 12.

Since the sides of a cube are all the same, the surface area of the cube is equal to 6 times the area of one face. For our dimensions, one face has an area of 12 * 12 or 144 in². Therefore, the total surface area is 6 * 144 = 864 in².

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Example Question #1 : Cubes

A room has dimensions of 18ft by 15ft by 9ft. The last dimension is the height of the room. It has one door that is 3ft by 7ft and two windows, each 2ft by 5ft. There is no trim to the floor, wall, doors, or windows. What is the total exposed wall space?

Possible Answers:

594 ft²

1134ft²

1093ft²

553 ft²

2389ft²

Correct answer:

553 ft²

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 9 ft high, we know 18 x 15 designates the area of the floor and ceiling. Based on this, we know that the room has the following dimensions for the walls: 18 x 9 and 15 x 9. 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 * (18 * 9 + 15 * 9) = 2 * (162 + 135) = 2 * 297 = 594 ft²

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

For the door: 3 * 7 = 21 ft²

For the windows: 2 * (2 * 5) = 20 ft²

The total wall space is therefore: 594 – 21 – 20 = 553 ft²

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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:

Correct answer:

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.

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

A certain cube has a side length of 25 m. How many square tiles, each with an area of 5 m², are needed to fully cover the surface of the cube?

Possible Answers:

750

500

100

1000

200

Correct answer:

750

Explanation:

A cube with a side length of 25m has a surface area of:

25m * 25m * 6 = 3,750 m²

(The surface area of a cube is equal to the area of one face of the cube multiplied by 6 sides. In other words, if the side of a cube is s, then the surface area of the cube is 6s^2.)

Each square tile has an area of 5 m².

Therefore, the total number of square tiles needed to fully cover the surface of the cube is:

3,750m²/5m²= 750

Note: the volume of a cube with side length s is equal to s³. Therefore, if asked how many mini-cubes with side length n are needed to fill the original cube, the answer would be:

s³/n³

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PSAT Math : Geometry

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #31 : Cubes

Example Question #31 : Solid Geometry

Example Question #1381 : Concepts

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

Example Question #221 : Geometry

Example Question #221 : Psat Mathematics

Example Question #3 : Solid Geometry

Example Question #1 : Cubes

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

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

All PSAT Math Resources

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