# ACT Math : Solid Geometry

## Example Questions

### Example Question #41 : Solid Geometry

Chemicals to clean a swimming pool cost \$0.24 per cubic foot of water. If a pool is 6 feet deep, 14 feet long and 8 feet wide, how much does it cost to clean the pool? Round to the nearest dollar.

Explanation:

The volume of the pool can be determined by multiplying the length, width, and height together.

Each cubit foot costs 24 cents, so:

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

A cube has edges that are three times as long as those of a smaller cube. The volume of the bigger cube is how many times larger than that of the smaller cube?

Explanation:

If we let  represent the length of an edge on the smaller cube, its volume is .

The larger cube has edges three times as long, so the length can be represented as . The volume is , which is .

The large cube's volume of  is 27 times as large as the small cube's volume of .

### Example Question #16 : How To Find The Volume Of A Cube

A tank measuring 3in wide by 5in deep is 10in tall.  If there are two cubes with 2in sides in the tank, how much water is needed to fill it?

Explanation:

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

What is the volume of a cube that has a side length of  inches?

Explanation:

We are given the side length of a cube so we simply plug that into the formula for the volume of a cube.

That formula is , and so the correct answer is

.

Make sure to check your units,  is the correct number, but the units are squared rather than cubed.

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

What is the volume of a cube with a side of length 1 cm?

Explanation:

The formula for the volume of a cube with a side of length  is:

### Example Question #3 : How To Find The Volume Of A Cube

A cube of sponge has volume . When water is added, the sponge triples in length along each dimension. What is the new volume of the cube, in cubic centimeters?

Explanation:

If our original cube has a volume of , then the length of one of its edges is . Triple each edge to , then cube the result, and we obtain

### Example Question #4 : How To Find The Volume Of A Cube

Find the volume of a cube with side length 10.

Explanation:

To find volume, simply cube the side length. Thus,

### Example Question #41 : Solid Geometry

Find the volume of a cube whose side length is 0.1.

Explanation:

To solve, simply use the formula for volume of a cube. The volume of a cube is length times width times height. Since the height, width, and length are equal on a cube the formula becomes side cubed.

We are given the side length of 0.1 thus,

### Example Question #42 : Solid Geometry

Find the volume of a cube given side length is .

Explanation:

To solve, simply cube the side length. Thus,

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

A square pyramid has a volume of  and a height of . What is the perimeter of the base of the pyramid?

Explanation:

The formula for the volume of a pyramid is:

We know that  and ; however, this still leaves us with two variables in the equation:  and . By definition, a square pyramid's base has sides of equal length, meaning that  and  are the same. Therefore, we can substitute  for , or .

This gives us a new equation of:

We then plug in the variables we know:

Multiply both sides by 3:

Divide both sides by 9:

Therefore, we now know that the length of the pyramid's base is . The question, however, asks for the perimeter of the pyramid's base. Since all of the sides of the base are the same, they must all be . So we multiply . Therefore, the perimeter of the pyramid's base is .