# GMAT Math : Calculating the surface area of a cube

## Example Questions

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### Example Question #1 : Calculating The Surface Area Of A Cube

What is the surface area of a box that is 3 feet long, 2 feet wide, and 4 feet high?

Explanation:

### Example Question #12 : Cubes

What is the surface area of a cube with side length 4?

Explanation:

### Example Question #2 : Calculating The Surface Area Of A Cube

The surface area of a certain cube is 150 square feet. If the width of the cube is increased by 2 feet, the length decreased by 2 feet and the height increased by 1 foot, what is the new surface area?

Explanation:

The first step to answering this qestion is to determine the original length of the sides of the cube. The surface area of a cube is given by:

Where  is the length of each side. This tells us that for our cube:

==>   ==>

If the width increases by 2, the length decreases by 2 and the height increases by 1:

, ,

We now have a rectangular prism. The surface area of a rectangular prism is given by:

For our prism:

### Example Question #14 : Cubes

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

Explanation:

### Example Question #15 : Cubes

A cube is inscribed inside a sphere with surface area . Give the volume of the cube.

Explanation:

Each diagonal of the inscribed cube is a diameter of the sphere, so its length is the sphere's diameter, or twice its radius.

The sphere has surface area , so the radius is calculated as follows:

The diameter of the circle - and the length of a diagonal of the cube - is twice this, or 10.

Now, let  be the length of one edge of the cube. By the three-dimensional extension of the Pythagorean Theorem,

The volume of the cube is the cube of this, or

### Example Question #2 : Calculating The Surface Area Of A Cube

A sphere of volume  is inscribed inside a cube. Give the surface area of the cube.

Explanation:

The diameter of a sphere is equal to the length of an edge of the cube in which it is inscribed. We can derive the radius using the volume formula:

Twice this, or 12, is the diameter, and, subsequently, the length of an edge of the cube. If , the surface area is

### Example Question #1 : Calculating The Surface Area Of A Cube

A cube is inscribed inside a sphere of volume . Give the surface area of the cube.

Explanation:

The diameter of a sphere is equal to the length of a diagonal of the cube it circumscribes. We can derive the radius using the volume formula:

Twice this, or 12, is the diameter, and, subsequently, the length of a diagonal of the cube. By an extension of the Pythagorean Theorem, if  is the length of an edge of the cube,

The surface area is six times this:

### Example Question #2 : Calculating The Surface Area Of A Cube

Cube A is inscribed inside a sphere, which is inscribed inside Cube B. Give the ratio of the surface area of Cube B to that of Cube A.

Explanation:

Suppose the sphere has diameter

Then Cube B, the circumscribing cube, has as its edge length the diameter , and its surface area is .

Also, Cube A, the inscribed cube, has this diameter as the length of its diagonal. If  is the length of an edge, then from the three-dimensional extension of the Pythagorean Theorem,

The surface area is , so

.

The ratio of the surface areas is

The correct choice is .

### Example Question #1 : Calculating The Surface Area Of A Cube

The length of one side of a cube is 4 meters. What is the surface area of the cube?

Explanation:

By definition, all sides of a cube are equal in length, so each face is a square, There are six faces on a cube, so its total surface area is six times the area of one of its square faces. If one of its sides is 4 meters, then this will also be the other dimension of one of its square faces, so the total surface area is:

### Example Question #4 : Calculating The Surface Area Of A Cube

Find the surface area of a cube whose side length is .