# ACT Math : Solid Geometry

## Example Questions

### Example Question #4 : Cubes

Given the volume of a cube is , find the side length.

Explanation:

To find side length, simply realize that volume is the side length cubed. Thus,

### Example Question #5 : Cubes

Find the length of the edge of a cube given the volume is .

Explanation:

To solve, simply take the cube root of the volume. Thus,

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

Find the length of the diagonal of a cube with side length of .

Explanation:

We begin with a picture, noting that the diagonal, labeled as , is the length across the cube from one vertex to the opposite side's vertex.

However, the trick to solving the problem is to also draw in the diagonal of the bottom face of the cube, which we labeled .

Note that this creates two right triangles.  Though our end goal is to find , we can begin by looking at the right triangle in the bottom face to find .  Using either the Pythagorean Theorem or the fact that we have a 45-45-90 right traingle, we can calculate the hypotenuse.

Now that we know the value of , we can turn to our second right triangle to find  using the Pythagorean Theorem.

Taking the square root of both sides and simplifying gives the answer.

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

What is the diagonal length for a cube with volume of  ?  Round to the nearest hundredth.

Explanation:

Recall that the volume of a cube is computed using the equation

, where  is the length of one side of the cube.

So, for our data, we know:

Using your calculator, take the cube root of both sides. You can always do this by raising  to the  power if your calculator does not have a varied-root button.

If you get , the value really should be rounded up to . This is because of calculator estimations. So, if the sides are  , you can find the diagonal by using a variation on the Pythagorean Theorem working for three dimensions:

This is . Round it to .

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

What is the length of the diagonal of a cube with a volume of ?

Explanation:

Recall that the diagonal of a cube is most easily found when you know that cube's dimensions. For the volume of a cube, the pertinent equation is:

, where  represents the length of one side of the cube. For our data, this gives us:

Now, you could factor this by hand or use your calculator. You will see that  is .

Now, we find the diagonal by using a three-dimensional version of the Pythagorean Theorem / distance formula:

or

You can rewrite this:

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

If the surface area of a cube equals 96, what is the length of one side of the cube?

3

6

5

4

4

Explanation:

The surface area of a cube = 6a2 where a is the length of the side of each edge of the cube. Put another way, since all sides of a cube are equal, a is just the lenght of one side of a cube.

We have 96 = 6a→ a2 = 16, so that's the area of one face of the cube.

Solving we get √16, so a = 4

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

The side of a cube has a length of . What is the total surface area of the cube?

Explanation:

A cube has 6 faces. The area of each face is found by squaring the length of the side.

Multiply the area of one face by the number of faces to get the total surface area of the cube.

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

What is the surface area of a cube if its height is 3 cm?

Explanation:

The area of one face is given by the length of a side squared.

The area of 6 faces is then given by six times the area of one face: 54 cm2.

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

A sphere with a volume of  is inscribed in a cube, as shown in the diagram below.

What is the surface area of the cube, in ?

Explanation:

We must first find the radius of the sphere in order to solve this problem. Since we already know the volume, we will use the volume formula to do this.

With the radius of the sphere in hand, we can now apply it to the cube. The radius of the sphere is half the distance from the top to the bottom of the cube (or half the distance from one side to another). Therefore, the radius represents half of a side length of a square. So in this case

The formula for the surface area of a cube is:

The surface area of the cube is

### Example Question #5 : How To Find The Surface Area Of A Cube

What is the surface area, in square inches, of a four-inch cube?