### All GRE Math Resources

## Example Questions

### Example Question #1 : Cubes

Quantity A: The length of a side of a cube with a volume of .

Quantity B: The length of a side of a cube with surface area of .

Which of the following is true?

**Possible Answers:**

Quantity B is larger.

Quantity A is larger.

The relationship between the two quantities cannot be determined.

The two quantities are equal

**Correct answer:**

The two quantities are equal

Recall that the equation for the volume of a cube is:

Since the sides of a cube are merely squares, the surface area equation is just times the area of one of those squares:

So, for our two quantities:

**Quantity A**

Use your calculator to estimate this value (since you will not have a square root key). This is .

**Quantity B**

First divide by :

Therefore,

Therefore, the two quantities are equal.

### Example Question #2 : Cubes

What is the length of an edge of a cube with a surface area of ?

**Possible Answers:**

**Correct answer:**

The surface area of a cube is made up of squares. Therefore, the equation is merely times the area of one of those squares. Since the sides of a square are equal, this is:

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

For our data, we know:

This means that:

Now, while you will not have a calculator with a square root key, you do know that . (You can always use your calculator to test values like this.) Therefore, we know that . This is the length of one side

### Example Question #3 : Cubes

If a cube has a total surface area of square inches, what is the length of one edge?

**Possible Answers:**

There is not enough information given.

**Correct answer:**

There are 6 sides to a cube. If the total surface area is 54 square inches, then each face must have an area of 9 square inches.

Every face of a cube is a square, so if the area is 9 square inches, each edge must be 3 inches.

### Example Question #12 : Solid Geometry

The surface area of a cube is 486 units. What is the distance of its diagonal (e.g. from its front-left-bottom corner to its rear-right-top corner)?

**Possible Answers:**

9√(3)

9

81

9√(2)

None of the others

**Correct answer:**

9√(3)

First, we must ascertain the length of each side. Based on our initial data, we know that the 6 faces of the cube will have a surface area of 6x^{2}. This yields the equation:

6x^{2} = 486, which simplifies to: x^{2} = 81; x = 9.

Therefore, each side has a length of 9. Imagine the cube is centered on the origin. This means its "front-left-bottom corner" will be at (–4.5, –4.5, 4.5) and its "rear-right-top corner" will be at (4.5, 4.5, –4.5). To find the distance between these, we use the three-dimensional distance formula:

d = √((x_{1} – x_{2})^{2} + (y_{1} – y_{2})^{2} + (z_{1} – z_{2})^{2})

For our data, this will be:

√( (–4.5 – 4.5)^{2} + (–4.5 – 4.5)^{2} + (4.5 + 4.5)^{2}) =

√( (–9)^{2} + (–9)^{2} + (9)^{2}) = √(81 + 81 + 81) = √(243) =

√(3 * 81) = √(3) * √(81) = 9√(3)

### Example Question #4 : Cubes

You have a rectangular box with dimensions 6 inches by 6 inches by 8 inches. What is the length of the shortest distance between two non-adjacent corners of the box?

**Possible Answers:**

**Correct answer:**

The shortest length between any two non-adjacent corners will be the diagonal of the smallest face of the rectangular box. The smallest face of the rectangular box is a six-inch by six-inch square. The diagonal of a six-inch square is .

### Example Question #5 : Cubes

What is the length of the diagonal of a cube with side lengths of each?

**Possible Answers:**

**Correct answer:**

The diagonal length of a cube is found by a form of the distance formula that is akin to the Pythagorean Theorem, though with an additional dimension added to it. It is:

, or , or

Now, if the the value of is , we get simply

### Example Question #6 : Cubes

What is the length of the diagonal of a cube that has a surface area of ?

**Possible Answers:**

**Correct answer:**

To begin, the best thing to do is to find the length of a side of the cube. This is done using the formula for the surface area of a cube. Recall that a cube is made up of squares. Therefore, its surface area is:

, where is the length of a side.

Therefore, for our data, we have:

Solving for , we get:

This means that

Now, the diagonal length of a cube is found by a form of the distance formula that is akin to the Pythagorean Theorem, though with an additional dimension added to it. It is:

, or , or

Now, if the the value of is , we get simply

### Example Question #21 : Solid Geometry

What is the volume of a rectangular box that is twice as long as it is high, and four times as wide as it is long?

**Possible Answers:**

5*L*

2*L*^{3}

2*L*^{2}

8

4*L*^{3}

**Correct answer:**

2*L*^{3}

The box is 2 times as long as it is high, so *H* = *L*/2. It is also 4 times as wide as it is long, so *W* = 4*L*. Now we need volume = *L* * *W* * *H* = *L* * 4*L* * *L*/2 = 2*L*^{3}.

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

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

**Possible Answers:**

**Correct answer:**

The surface area of a cube is merely the sum of the surface areas of the squares that make up its faces. Therefore, the surface area equation understandably is:

, where is the side length of any one side of the cube. For our values, we know:

Solving for , we get:

or

Now, the volume of a cube is defined by the simple equation:

For , this is:

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

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:

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