# ACT Math : How to find the volume of a polyhedron

## Example Questions

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### Example Question #1 : Solid Geometry

Roberto has a swimming pool that is in the shape of a rectangular prism. His swimming pool is  meters wide,  meters long, and  meters deep. He needs to fill up the pool for summer, and his hose fills at a rate of  cubic meters per hour. How many hours will it take for Roberto to fill up the swimming pool?

Explanation:

First, find the volume of the pool. For a rectangular prism, the formula for the volume is the following:

For the swimming pool,

cubic meters

Now, because the hose only fills up  cubic meters per hour, divide the total volume by  to find how long it will take for the pool to fill.

It will take the pool  hours to fill by hose.

### Example Question #3 : Other Polyhedrons

Matt baked a rectangular cake for his mom's birthday. The cake was  inches long,  inches wide, and  inches high. If he cuts the cake into pieces that are  inches long,  inches wide, and  inches high, how many pieces of cake can he cut?

Explanation:

First, find the volume of the cake. For a rectangular prism,

Next, find the volume of each individual slice.

Now, divide the volume of the entire cake by the volume of the slice to get how many pieces of cake Matt can cut.

### Example Question #4 : Other Polyhedrons

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

The volume will be  times greater.

The volume will be  times greater.

The volume will be  times greater.

The volume will be  times greater.

The volume will be  times greater.

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 greater than the original.

### Example Question #5 : Other Polyhedrons

A cube has a surface area of . What is its volume?

Explanation:

First, find the side lengths of 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, recall that the volume of a cube is given by the following equation:

### Example Question #6 : Other Polyhedrons

The volume of the right triangular prism is . Find the value of .

Explanation:

The volume of a right triangular prism is given by the following equation:

Now, for the given question, the height is .

Since the area of the base is a right triangle, we can plug in the given values to find .

### Example Question #7 : Other Polyhedrons

The tent shown below is in the shape of a triangular prism. What is the volume of this tent in cubic feet?

Explanation:

The volume of a right triangular prism is given by the following equation:

### Example Question #2 : Solid Geometry

The height of a box is twice its width and half its length. If the volume of the box is , what is the length of the box?

Explanation:

For a rectangular prism, the formula for the volume is the following:

Now, we know that the height is twice its width. We can rewrite that as:

We also know that the height is half its length. That can be written as:

Now, we can plug in the values of the length, width, and height in terms of height to find the height.

The question wants to find the length of the box. Plug in the value of the height in the earlier equation we wrote earlier to represent the relatioinship between the height and the length.

### Example Question #3 : Solid Geometry

In cubic inches, find the volume of a tetrahedron that has a surface area of .

Explanation:

First, we will need to find the length of a side of the tetrahedron.

We can use the surface area to find the lengh of a side. Recall that the formula to find the surface of a tetrahedron:

, where  is the side length.

Now, recall the formula to find the volume of a tetrahedron:

### Example Question #10 : Other Polyhedrons

Susan bought a chocolate bar that came in a container shaped like a triangular prism shown below. If the container is completely filled with chocolate, in cubic inches, what volume of chocoate did Susan buy?

Explanation:

The volume of a right triangular prism is given by the following equation:

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

Troy's company manufactures dice that are shaped like cubes and have side lengths of . If the plastic needed to make the dice costs  per cubic centimeter, how much does it cost Troy to make one die?