### All GMAT Math Resources

## Example Questions

### 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.

**Possible Answers:**

**Correct answer:**

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 #2 : 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?

**Possible Answers:**

**Correct answer:**

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 .

**Possible Answers:**

**Correct answer:**

To solve, remember that the equation for surface area of a cube is:

### Example Question #21 : Cubes

Aperture labs makes a variety of cubes. If each cube has a volume of , what is the surface area of the cube?

**Possible Answers:**

**Correct answer:**

**Aperture labs makes a variety of cubes. If each cube has a volume of , what is the surface area of the cube?**

Let's work backwards from our goal in this question.

We know that we need to find surface area. To find surface area of a cube, we can use the following equation:

Where l is the length of one side.

Next, let's look at the volume formula:

So, we can find our length

Let's leave l like that for the moment, and use it to find our surface area.

### Example Question #1 : Calculating The Volume Of A Cube

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

**Possible Answers:**

**Correct answer:**

### Example Question #21 : Cubes

The length, width, and height of a rectangular prism, in inches, are three different prime numbers. All three dimensions are between six feet and seven feet. What is the volume of the prism?

**Possible Answers:**

It is impossible to tell from the information given.

**Correct answer:**

Six feet and seven feet are equal to, respectively, 72 inches and 84 inches. There are three different prime numbers between 72 and 84 - 73, 79, and 83 - so these are the three dimensions of the prism in inches. The volume of the prism is

cubic inches.

### Example Question #21 : Cubes

The length of a diagonal of one face of a cube is . Give the volume of the cube.

**Possible Answers:**

The correct answer is not among the other responses.

**Correct answer:**

A diagonal of a square has length times that of a side, so each side of each square face of the cube has length . Cube this to get the volume:

### Example Question #2 : Calculating The Volume Of A Cube

The length of a diagonal of a cube is . Give the volume of the cube.

**Possible Answers:**

**Correct answer:**

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

Cube the sidelength to get the volume:

### Example Question #21 : Cubes

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

**Possible Answers:**

**Correct answer:**

The sidelength of the cube is the diameter of the inscribed sphere, which is twice that sphere's radius. The sphere has surface area , so the radius is calculated as follows:

The diameter of the sphere - and the sidelength of the cube - is twice this, or .

Cube this sidelength to get the volume of the cube:

### Example Question #111 : Rectangular Solids & Cylinders

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

**Possible Answers:**

The correct answer is not given among the other responses.

**Correct answer:**

The diameter of the circle - twice its radius - coincides with the length of a diagonal of the inscribed cube. The sphere has volume , so the radius is calculated as follows:

The diameter of the sphere - and the length of a diagonal of the cube - is twice this, or 6.

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

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