# ACT Math : Solid Geometry

## Example Questions

### Example Question #10 : How To Find The Volume Of A Polyhedron

A pyramid is placed inside a cube so that they share a base and height. If the surface area of the cube is , what is the volume of the pyramid, in square feet?

Explanation:

First, we need to find the length of a side for the cube.

Recall that the surface area of the cube is given by the following equation:

, where  is the length of a side.

Plugging in the surface area given by the equation, we can then find the side length of the cube.

Now, because the pyramid and the cube share a base, we know that the pyramid must be a square pryamid.

Recall how to find the volume of a pyramid:

Now, since the pyramid is the same height as the cube, the height of the pyramid is also .

### Example Question #11 : Solid Geometry

In cubic feet, find the volume of the pentagonal prism illustrated below. The pentagon has an area of , and the prism has a height of .

Explanation:

For any prism, the volume is given by the following equation:

The question gives us the area of the base and the height.

### Example Question #12 : Solid Geometry

If the side lengths of a cube are tripled, what effect will it have on the volume?

The volume will be  times as large.

The volume will be  times as large.

The volume will be  times as large.

The volume will be  times as large.

The volume will be  times as large.

Explanation:

Start by taking a cube that is . The volume of this cube is .

Next, triple the sides of this cube so that it becomes . The volume of this cube is

The volume of the new cube is  times as large as the original.

### Example Question #13 : Solid Geometry

The surface area of a cylinder is given by , where  is the radius and  is the height.  If a cylinder has a surface area of   and a height of  , what is its radius?

Explanation:

The fastest way to solve this problem is to plug all of the answer choices in for  and look for an output of .

Alternatively, one could set the surface area formula equal to  (knowing that ), but this would require solving a quadratic.

Rearranging this we get

Now factoring out a  we get:

Since we cannot have a negative length our answer is .

### Example Question #14 : Solid Geometry

If a half cylinder with a height of 5 and semicircular bases with a radius of 2, what is the surface area?

Explanation:

The surface area of the half cylinder will consist of the lateral area and the base area.

There are three parts to the surface area:

Rectangular region, semi-circular bases of the half cylinder, and outer face of the cylinder.

The cross section of the half cylinder is a rectangle.  Find the area of the rectangle.  The diameter of the semicircle represents the width of the rectangle, which is double the radius.  The length of the rectangle is the height of the cylinder.

Next, find the area of a semicircular bases.

Since there are two semicircular bases of the semi-cylinder, the total area of the semicircular bases is .

Find the area of the outer region.   The area of the outer region is the half circumference multiplied by the height of the cylinder.

Sum the areas of the rectangle, the two circular bases, and the outer region to find the lateral area.

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

Our backyard pool holds 10,000 gallons.  Its average depth is 4 feet deep and it is 10 feet long.  If there are 7.48 gallons in a cubic foot, how wide is the pool?

33  ft

133 ft

100 ft

30 ft

7.48 ft

33  ft

Explanation:

There are 7.48 gallons in cubic foot. Set up a ratio:

1 ft3 / 7.48 gallons = x cubic feet / 10,000 gallons

Pool Volume = 10,000 gallons = 10,000 gallons * (1 ft3/ 7.48 gallons) = 1336.9 ft3

Pool Volume = 4ft x 10 ft x WIDTH = 1336.9 cubic feet

Solve for WIDTH:

4 ft x 10 ft x WIDTH = 1336.9 cubic feet

WIDTH = 1336.9 / (4 x 10) = 33.4 ft

### Example Question #2 : How To Find The Length Of An Edge Of A Cube

A cube has a volume of 64cm3. What is the area of one side of the cube?

16cm3

16cm

4cm2

4cm

16cm2

16cm2

Explanation:

The cube has a volume of 64cm3, making the length of one edge 4cm (4 * 4 * 4 = 64).

So the area of one side is 4 * 4 = 16cm2

### Example Question #1 : Cubes

A cube as the volume of .

Find the length of a side of this cube.

Explanation:

The formula to find the volume of the cube is

Since we know the volume, we can set up the equation

### Example Question #2 : Cubes

A cube has a surface area of , what is the length of the side of the cube? (If necessary, round to the nearest hundredth.)

Explanation:

To find the length of the side of a square given the surface area, use the surface area formula and solve for :

, now divide both sides by 6
, now square root both sides
.

### Example Question #3 : Cubes

A certain cubic box when unfolded and laid flat on a table covers exactly  square units of space. What is the width of the box, in units?