# ACT Math : Solid Geometry

## Example Questions

### Example Question #3 : Prisms

Sturgis is in charge of designing a new exhibit in the shape of a rectangular prism for a local aquarium. The exhibit will hold alligator snapping turtles and needs to have a volume of . Sturgis knows that the exhibit will be  long and go  back into the wall.

What will the height of the new exhibit be?

Explanation:

This sounds like a geometry problem, so start by drawing a picture so that you know exactly what you are dealing with.

Because we are dealing with rectangular prisms and volume, we will need the following formula:

Or

We are solving for height, so you can begin by rearranging the equation to get  by itself:

Then, plug in our knowns (,  and )

Here is the problem worked out with a corresponding picture:

### Example Question #4 : Prisms

Sturgis is in charge of designing a new exhibit in the shape of a rectangular prism for a local aquarium. The exhibit will hold alligator snapping turtles and needs to have a volume of . Sturgis knows that the exhibit will be  long and go  back into the wall.

If three-quarters of the exhibit's volume will be water, how high up the wall will the water come?

Cannot be determined with the information provided

Explanation:

The trickiest part of this question is the wording. This problem is asking for the height of the water in the exhibit if the exhibit is three-quarters full. We can find this at least two different ways.

1) The longer way requires that we begin by finding three quarters of the total volume:

Now we go back to our volume equation, and since we are again looking for height, we want it solved for :

Becomes

2) The easier way requires that we recognize a key detail. If we take three-quarters of the volume without changing our length or width, our new height will just be three-quarters of the total height. We can solve for the total height of the exhibit by using the volume equation and rearranging it to solve for :

At this point, we can substitute in our given values and solve for :

So, the total height of the exhibit is . We can now easily solve for three-quarters of the total height:

### Example Question #5 : Prisms

A right, rectangular prism has has a length of , a width of , and a height of . What is the length of the diagonal of the prism?

Explanation:

First we must find the diagonal of the prism's base (). This can be done by using the Pythagorean Theorem with the length () and width ():

Therefore, the diagonal of the prism's base is . We can then use this again in the Pythagorean Theorem, along with the height of the prism (), to find the diagonal of the prism ():

Therefore, the length of the prism's diagonal is .

### Example Question #6 : Prisms

What is the diagonal of a rectangular prism with a height of 4, width of 4 and height of 6?

Cannot be determined

Explanation:

In order to solve this problem, it's helpful to visualize where the diagonal is within the prism.

In this image, the diagonal is the pink line. By noting how it relates to the blue and green lines, we can observe how the pink line is connected and creates a right triangle. This very quickly becomes a problem that employs the Pythagorean theorem.

The goal is essentially to find the hypotenuse of this sketched-in right triangle; however, only one of the legs is given: the green line, the height of the prism. The blue line can be solved for by understanding that it is the measurement of the diagonal of a 4x4 square.

Either using trig functions or the rules for a special 45/45/90 triangle, the blue line measures out to be

The rules for a 45/45/90 triangle: both legs are "" and the hypotenuse is "". Keep in mind, this is is only for isosceles right triangles.

Now that both legs are known, we can solve for the hypotenuse (diagonal).

### Example Question #7 : Prisms

Find the diagonal of a right rectangular prism if the length, width, and height are 3,4, and 5, respectively.

Explanation:

Write the diagonal formula for a rectangular prism.

Substitute and solve for the diagonal.

### Example Question #8 : Prisms

If the dimensions of a right rectangular prism are 1 yard by 1 foot by 1 inch, what is the diagonal in feet?

Explanation:

Convert the dimensions into feet.

The new dimensions of rectangular prism in feet are:

Write the formula for the diagonal of a right rectangular prism and substitute.

### Example Question #9 : Prisms

David wants to paint the walls in his bedroom. The floor is covered by a  carpet.  The ceiling is  tall. He selects a paint that will cover  per quart and  per gallon. How much paint should he buy?

2 gallons and 1 quart

1 gallon and 1 quart

1 gallon

3 quarts

1 gallon and 2 quarts

1 gallon and 2 quarts

Explanation:

Find the surface area of the walls: SAwalls = 2lh + 2wh, where the height is 8 ft, the width is 10 ft, and the length is 16 ft.

This gives a total surface area of 416 ft2. One gallon covers 300 ft2, and each quart covers 75 ft2, so we need 1 gallon and 2 quarts of paint to cover the walls.

### Example Question #10 : Prisms

A box is 5 inches long, 5 inches wide, and 4 inches tall. What is the surface area of the box?

Explanation:

The box will have six total faces: an identical "top and bottom," and identical "left and right," and an identical "front and back." The total surface area will be the sum of these faces.

Since the six faces consider of three sets of pairs, we can set up the equation as:

Each of these faces will correspond to one pair of dimensions. Multiply the pair to get the area of the face.

Substitute the values from the question to solve.

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

What is the surface area of a rectangular brick with a length of 12 in, a width of 8 in, and a height of 6 in?

None of the answers are correct

Explanation:

The formula for the surface area of a rectangular prism is given by:

SA = 2LW + 2WH + 2HL

SA = 2(12 * 8) + 2(8 * 6) + 2(6 * 12)

SA = 2(96) + 2(48) + 2(72)

SA = 192 + 96 + 144

SA = 432 in2

216 in2  is the wrong answer because it is off by a factor of 2

576 in3 is actually the volume, V = L * W * H

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

A box's length is twice as long as its width. Its height is the sum of its length and its width. What is the volume of this box if its length is 10 units?

units cubed

units cubed

units cubed

units cubed

units cubed