# HSPT Math : Geometry

## Example Questions

### 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?

1093ft2

553 ft2

1134ft2

2389ft2

594 ft2

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

### Example Question #592 : Problem Solving

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.

14

18

13

15

11

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 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?

500

750

200

1000

100

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 6s2.)

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:

3,750m2/5m= 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

### 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 ft3, what is the surface area of the cube?

513.375 in2

486 in2

69,984 in2

726 in2

73,926 in2

73,926 in2

Explanation:

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)πr2h

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

486π  = (1/3)πr* 2r = (2/3)πr3

Multiply both sides by (3/2π): 729 = r3

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 * s2, or for our data, A = 6 * 1112 = 73,926 in2

### Example Question #601 : Problem Solving

Angie is painting a 2 foot cube for a play she is in. She needs  of paint for every square foot she paints. How much paint does she need?

It is impossible to convert between metric units and feet.

Explanation:

First we must calculate the surface area of the cube. We know that there are six surfaces and each surface has the same area:

Now we will determine the amount of paint needed

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

A spherical orange fits snugly inside a small cubical box such that each of the six walls of the box just barely touches the surface of the orange.  If the volume of the box is 64 cubic inches, what is the surface area of the orange in square inches?

16π

64π

128π

32π

256π

16π

Explanation:

The volume of a cube is found by V = s3.  Since V = 64, s = 4.  The side of the cube is the same as the diameter of the sphere.  Since d = 4, r = 2.  The surface area of a sphere is found by SA = 4π(r2) = 4π(22) = 16π.

### Example Question #601 : Problem Solving

Steve's bedroom measures 20' by 18' by 8' high. He wants to paint the ceiling and all four walls using a paint that gets 360 square feet of coverage per gallon. A one-gallon can of the paint Steve wants costs $36.00; a one-quart can costs$13.00. What is the least amount of money that Steve can expect to spend on paint in order to paint his room?

Explanation:

Two of the walls have area ; two have area ; the ceiling has area

Therefore, the total area Steve wants to cover is

Divide 968 by 360 to get the number of gallons Steve needs to paint his bedroom:

Since , Steve needs to purchase either two gallon cans and three quart cans, or three gallon cans.

The first option will cost him ; the second option will cost him . The latter is the more economical option.

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

Give the surface area of the above box in square centimeters.

Explanation:

100 centimeters make one meter, so convert each of the dimensions of the box by multiplying by 100.

centimeters

centimeters

Use the surface area formula, substituting :

square centimeters

### Example Question #601 : Problem Solving

Note: Figure NOT drawn to scale.

Refer to the above diagram, which shows a square. Give the ratio of the area of the yellow region to that of the white region.

The correct answer is not given among the other choices.

Explanation:

The area of the entire square is the square of the length of a side, or

.

The area of the right triangle is half the product of its legs, or

.

The area of the yellow region is therefore the difference of the two, or

.

The ratio of the area of the yellow region to that of the white region is

; that is, 55 to 9.

### Example Question #241 : Geometry

The above depicts a rectangular swimming pool for an apartment. The pool is five feet deep everywhere.

An apartment manager wants to paint the four sides and the bottom of the swimming pool. One one-gallon can of the paint he wants to use covers  square feet. How many cans of the paint will the manager need to buy?

Explanation:

The bottom of the swimming pool has area

square feet.

There are two sides whose area is

square feet,

and two sides whose area is

square feet.