### All PSAT Math Resources

## Example Questions

### Example Question #1 : Solid Geometry

What is the surface area of a cube with a volume of 1728 in^{3}?

**Possible Answers:**

1728 in^{2}

144 in^{2}

72 in^{2}

432 in^{2}

864 in^{2}

**Correct answer:**

864 in^{2}

This problem is relatively simple. We know that the volume of a cube is equal to s^{3}, where s is the length of a given side of the cube. Therefore, to find our dimensions, we merely have to solve s^{3} = 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 in^{2}. Therefore, the total surface area is 6 * 144 = 864 in^{2}.

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

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:**

2389ft^{2}

553 ft^{2}

594 ft^{2}

1134ft^{2}

1093ft^{2}

**Correct answer:**

553 ft^{2}

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 ft^{2}

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

For the door: 3 * 7 = 21 ft^{2}

For the windows: 2 * (2 * 5) = 20 ft^{2}

The total wall space is therefore: 594 – 21 – 20 = 553 ft^{2}

### Example Question #1 : 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 ft^{2} of surface area. How many cans of paint must be bought to paint the walls of the room.

**Possible Answers:**

15

13

11

14

18

**Correct answer:**

14

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 ft^{2}

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

For the door: 2.5 * 8 = 17.5 ft^{2}

For the windows: 3 * 6 = 18 ft^{2}

The total wall space is therefore: 800 – 17.5 – 18 = 764.5 ft^{2}

Now, if one can of paint covers 57 ft^{2}, we calculate the number of cans necessary by dividing the total exposed area by 57: 764.5/57 = (approx.) 13.41.

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

### 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 m^{2}, are needed to fully cover the surface of the cube?

**Possible Answers:**

750

200

1000

100

500

**Correct answer:**

750

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

25m * 25m * 6 = 3,750 m^{2}

(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 6*s*^{2.})

Each square tile has an area of 5 m^{2}.

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

3,750m^{2}/5m^{2 }= 750

Note: the volume of a cube with side length *s* is equal to *s*^{3}. Therefore, if asked how many mini-cubes with side length *n* are needed to fill the original cube, the answer would be:

s^{3}/n^{3}

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

A company wants to build a cubical room around a cone so that the cone's height and diameter are 3 inch less than the dimensions of the cube. If the volume of the cone is 486π ft^{3}, what is the surface area of the cube?

**Possible Answers:**

73,926 in^{2}

486 in^{2}

513.375 in^{2}

69,984 in^{2}

726 in^{2}

**Correct answer:**

73,926 in^{2}

To begin, we need to solve for the dimensions of the cone.

The basic form for the volume of a cone is: *V* = (1/3)*πr*^{2}*h*

Using our data, we know that *h* = *2r* because the height of the cone matches its diameter (based on the prompt).

486*π* = (1/3)*πr*^{2 }* 2*r* = (2/3)*πr*^{3}

Multiply both sides by (3/2*π*): 729 = *r*^{3}

Take the cube root of both sides: r = 9

Note that this is **in feet**. The answers are in square inches. Therefore, convert your units to inches: 9 * 12 = 108, then add 3 inches to this: 108 + 3 = 111 inches.

The surface area of the cube is defined by: *A* = 6 * *s*^{2}, or for our data, *A* = 6 * 111^{2} = 73,926 in^{2}

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

If the volume of a cube is 216 cubic units, then what is its surface area in square units?

**Possible Answers:**

108

64

36

54

216

**Correct answer:**

216

The volume of a cube is given by the formula *V* = , where *V* is the volume, and *s* is the length of each side. We can set *V* to 216 and then solve for *s*.

In order to find *s*, we can find the cube root of both sides of the equaton. Finding the cube root of a number is the same as raising that number to the one-third power.

This means the length of the side of the cube is 6. We can use this information to find the surface area of the cube, which is equal to . The formula for surface area comes from the fact that each face of the cube has an area of , and there are 6 faces in a cube.

surface area =

The surface area of the square is 216 square units.

The answer is 216.

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

You have a cube with sides of 4.5 inches. What is the surface area of the cube?

**Possible Answers:**

**Correct answer:**

The area of one side of the cube is:

A cube has 6 sides, so the total surface area of the cube is

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

A cube has a surface area of 24. If we double the height of the cube, what is the volume of the new rectangular box?

**Possible Answers:**

**Correct answer:**

We have a cube with a surface area of 24, which means each side has an area of 4. Therefore, the length of each side is 2. If we double the height, the volume becomes .

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

A cube has a surface area of 10m^{2}. If a cube's sides all double in length, what is the new surface area?

**Possible Answers:**

40m^{2}

640m^{2}

80m^{2}

320m^{2}

20m^{2}

**Correct answer:**

40m^{2}

The equation for surface area of the original cube is 6s^{2}. If the sides all double in length the new equation is 6(2s)^{2} or 6 * 4s^{2}. This makes the new surface area 4x that of the old. 4x10 = 40m^{2}

### Example Question #61 : Solid Geometry

If a cube has an area of , then what is the surface area of this cube?

**Possible Answers:**

**Correct answer:**

The area of a cube is equal to the measure of one edge cubed. If we take the cube root of , we get , so the edges of this cube measure ; therefore, one face of the cube has an area of square inches, because .

There are sides to a cube, so

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