GRE Math : Cubes

Study concepts, example questions & explanations for GRE Math

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Example Questions

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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

Explanation:

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:

Explanation:

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:

 

Explanation:

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)

Explanation:

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 6x2. This yields the equation:

6x2 = 486, which simplifies to: x2 = 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 = √((x1 – x2)2 + (y1 – y2)2 + (z1 – z2)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:

\dpi{100} \small 6

\dpi{100} \small 8

\dpi{100} \small 4\sqrt{3}

\dpi{100} \small 8\sqrt{2}

\dpi{100} \small 6\sqrt{2}

Correct answer:

\dpi{100} \small 6\sqrt{2}

Explanation:

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 \dpi{100} \small 6\sqrt{2}.

Example Question #5 : Cubes

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

Possible Answers:

Correct answer:

Explanation:

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:

Explanation:

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:

5L

2L3

2L2

8

4L3

Correct answer:

2L3

Explanation:

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 = 4L. Now we need volume = L * W * H = L * 4L * L/2 = 2L3.

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:

Explanation:

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:

Explanation:

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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