### All GRE Math Resources

## Example Questions

### Example Question #24 : Solid Geometry

The volume of a cube is . If the side length of this cube is tripled, what is the new volume?

**Possible Answers:**

**Correct answer:**

Recall that the volume of a cube is defined by the equation:

, where is the side length of the cube.

Therefore, if we know that , we can solve:

This means that .

Now, if we triple to , the new volume of our cube will be:

### Example Question #331 : Geometry

What is the volume of a cube with surface area of ?

**Possible Answers:**

**Correct answer:**

Recall that the equation for the surface area of a cube is merely derived from the fact that the cube's faces are made up of squares. It is therefore:

For our values, this is:

Solving for , we get:

, so

Now, the volume of a cube is merely:

Therefore, for , this value is:

### Example Question #332 : Geometry

A cube has a volume of 64, what would it be if you doubled its side lengths?

**Possible Answers:**

**Correct answer:**

To find the volume of a cube, you multiple your side length 3 times (s*s*s).

To find the side length from the volume, you find the cube root which gives you 4

.

Doubling the side gives you 8

.

The volume of the new cube would then be 512

.

### Example Question #331 : Geometry

A rectangular box is 6 feet wide, 3 feet long, and 2 feet high. What is the surface area of this box?

**Possible Answers:**

64 sq ft

36 sq ft

22 sq ft

48 sq ft

72 sq ft

**Correct answer:**

72 sq ft

The surface area formula we need to solve this is 2*ab* + 2*bc* + 2*ac*. So if we let *a* = 6, *b* = 3, and *c* = 2, then surface area:

= 2(6)(3) + 2(3)(2) +2(6)(2)

= 72 sq ft.

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

What is the surface area of a rectangular box that is 3 feet high, 6 feet long, and 4 feet wide?

**Possible Answers:**

108

144

72

84

96

**Correct answer:**

108

Surface area of a rectangular solid

= 2*lw* + 2*lh* + 2*wh*

= 2(6)(4) + 2(6)(3) + 2(4)(3)

= 108

### Example Question #332 : Geometry

A large cube is made by fitting 8 smaller, identical cubes together. If the volume of each of the smaller cubes is 27, what is the surface area of the large cube?

**Possible Answers:**

**Correct answer:**

Since the volume of the smaller cubes with edges, , is 27, we have:

.

The large cube has edges .

So the surface area of the large cube is:

.

### Example Question #333 : Geometry

**Quantity A:**

The surface area of a cube with a volume of .

**Quantity B:**

The volume of a cube with a surface area of .

**Possible Answers:**

The relationshp cannot be determined from the information given.

Both quantities are equal.

Quantity A is greater.

Quantity B is greater.

**Correct answer:**

Quantity B is greater.

The relationship can be determined, because it is possible to find the surface area of a cube from the volume and vice versa.

Quantity A:

To find the surface area of the cube, you must find the side length. To find the side length from the volume, you must find the cube root.

Find the cube root of the volume.

Insert into surface area equation.

Quantity B:

To find the volume of a cube, you must find the side length. To find the side length from the surface area, you must divide by 6, then find the square root of the result. Then, cube that result.

Divide by 6.

Square root.

Now, to find the volume.

Quantity B is greater.

### Example Question #1 : How To Find The Diameter Of A Sphere

A cube with a surface area of 216 square units has a side length that is equal to the diameter of a certain sphere. What is the surface area of the sphere?

**Possible Answers:**

**Correct answer:**

Begin by solving for the length of one side of the cube. Use the formula for surface area to do this:

s= length of one side of the cube

The length of the side of the cube is equal to the diameter of the sphere. Therefore, the radius of the sphere is 3. Now use the formula for the surface area of a sphere:

The surface area of the sphere is .

### Example Question #32 : Solid Geometry

The surface area of a sphere is . What is its diameter?

**Possible Answers:**

**Correct answer:**

The surface area of a sphere is defined by the equation:

For our data, this means:

Solving for , we get:

or

The diameter of the sphere is .

### Example Question #33 : Solid Geometry

The volume of one sphere is . What is the diameter of a sphere of half that volume?

**Possible Answers:**

**Correct answer:**

Do not assume that the diameter will be half of the diameter of a sphere with volume of* *. Instead, begin with the sphere with a volume of* . *Such a simple action will prevent a vexing error!

Thus, we know from our equation for the volume of a sphere that:

Solving for , we get:

If you take the cube-root of both sides, you have:

First, you can factor out an :

Next, factor the :

Which simplifies to:

Thus, the diameter is double that or: