# High School Math : Solid Geometry

## Example Questions

### Example Question #1 : Cubes

Find the length of an edge of the following cube:

The volume of the cube is .

Explanation:

The formula for the volume of a cube is

,

where  is the length of the edge of a cube.

Plugging in our values, we get:

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

Find the length of an edge of the following cube:

The volume of the cube is .

Explanation:

The formula for the volume of a cube is

,

where  is the length of the edge of a cube.

Plugging in our values, we get:

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

What is the length of an edge of a cube that has a surface area of 54?

Explanation:

The surface area of a cube can be determined using the following equation:

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

Find the length of the diagonal connecting opposite corners of a cube with sides of length .

units

units

units

units

units

Explanation:

Find the diagonal of one face of the cube using the Pythagorean Theorem applied to a triangle formed by two sides of that face ( and ) and the diagonal itself ():

This diagonal is now the base of a new right triangle (call this ). The height of that triangle is an edge of the cube that runs perpendicular to this diagonal (call this ). The third side of the triangle formed by  and  is a line from one corner of the cube to the other, i.e., the cube's diagonal (call this ). Use the Pythagorean Theorem again with the triangle formed by , and  to find the length of this diagonal.

### Example Question #31 : Solid Geometry

Find the length of the diagonal of the following cube:

Explanation:

To find the length of the diagonal, use the formula for a  triangle:

The length of the diagonal is .

### Example Question #1 : Cubes

What is the surface area of a cube with a side length of 15?

Explanation:

To find the surface area of a cube we must count the number of surface faces and add the areas of each of them together.

In a cube there are 6 faces, each a square with the same side lengths.

In this example the side lengths is 15 so the area of each square would be

We then multiply this number by 6, the number of faces of the cube, to get

Our answer for the surface area is .

### Example Question #571 : Geometry

What is the surface area of a cube with a side length of ?

Explanation:

To find the surface area of a cube, we must count the number of surface faces and add the areas of each together. In a cube there are  faces, each a square with the same side lengths. In this example the side length is .

The area of a square is given by the equation . Using our side length, we can solve the area of once face of the cube.

We then multiply this number by , the number of faces of the cube to find the total surface area.

Our answer for the surface area is .

### 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

4

5

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

What is the surface area of a cube with a side length of ?

Explanation:

In order to find the surface area of a cube, use the formula .

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

What is the surface area, in inches, of a rectangular prism with a length of , a width of , and a height of ?

Explanation:

In order to find the surface area of a rectangular prism, use the formula .

However, all units must be the same. All of the units of this problem are in inches with the exception of .

Convert to inches.

Now, we can insert the known values into the surface area formula in order to calulate the surface area of the rectangular prism.

If you calculated the surface area to equal , then you utilized the volume formula of a rectangular prism, which is ; this is incorrect.