GRE Math : Solid Geometry

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Example Questions

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

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:

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

$\dpi{100} \small 8$

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

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

$\dpi{100} \small 6$

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

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

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

Possible Answers:

$4\sqrt{3}\:in$

$3\sqrt{2}\:in$

$14\sqrt{3}\:in$

$12\:in$

$\sqrt{15}\:in$

Correct answer:

$4\sqrt{3}\:in$

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:

$\sqrt{s^2+s^2+s^2}$ , or $\sqrt{3s^2}$ , or $s\sqrt{3}$

Now, if the the value of is , we get simply $4\sqrt{3}$

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Example Question #3 : How To Find The Diagonal Of A Cube

What is the length of the diagonal of a cube that has a surface area of 726 in^2 ?

Possible Answers:

$11\sqrt{3}\:in$

$11\:in$

$12\sqrt{2}\:in$

$22\:in$

$12\:in$

Correct answer:

$11\sqrt{3}\:in$

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:

SA=6s^2 , where is the length of a side.

Therefore, for our data, we have:

726=6s^2

Solving for , we get:

121=s^2

This means that s=11

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:

$\sqrt{s^2+s^2+s^2}$ , or $\sqrt{3s^2}$ , or $s\sqrt{3}$

Now, if the the value of is , we get simply $11\sqrt{3}$

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

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:

4L³

2L²

2L³

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.

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

What is the volume of a cube with a surface area of 150 in^2 ?

Possible Answers:

$125\:in^3$

$150\:in^3$

$25\:in^3$

$5\:in^3$

$325\:in^3$

Correct answer:

$125\:in^3$

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:

SA = 6*s^2 , where is the side length of any one side of the cube. For our values, we know:

150=6s^2

Solving for , we get:

25=s^2 or s=5

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

V=s^3

For s=5 , this is:

$V=5^3=125\:in^3$

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Example Question #24 : Solid Geometry

The volume of a cube is $64\:in^3$ . If the side length of this cube is tripled, what is the new volume?

Possible Answers:

$192\:in^3$

$768\:in^3$

$1728\:in^3$

$924\:in^3$

$1024\:in^3$

Correct answer:

$1728\:in^3$

Explanation:

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

V=s^3 , where is the side length of the cube.

Therefore, if we know that V=64 , we can solve:

64=s^3

This means that s=4 .

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

$V=12^3 = 1728\:in^3$

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

What is the volume of a cube with surface area of $384\:in^2$ ?

Possible Answers:

$512\:in^3$

$64\:in^3$

$984\:in^3$

$224\:in^3$

$256\:in^3$

Correct answer:

$512\:in^3$

Explanation:

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:

SA=6s^2

For our values, this is:

384=6s^2

Solving for , we get:

64=s^2 , so s=8

Now, the volume of a cube is merely:

V=s^3

Therefore, for s=8 , this value is:

$V=8^3=512\:in^3$

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Example Question #3 : How To Find The Volume Of A Cube

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

Possible Answers:

500

128

512

256

Correct answer:

512

Explanation:

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

$(\sqrt[3]{64})$ .

Doubling the side gives you 8

(4*2=8) .

The volume of the new cube would then be 512

(8*8*8=512) .

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

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

Possible Answers:

72 sq ft

36 sq ft

64 sq ft

48 sq ft

22 sq ft

Correct answer:

72 sq ft

Explanation:

The surface area formula we need to solve this is 2ab + 2bc + 2ac. 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.

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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

Correct answer:

108

Explanation:

Surface area of a rectangular solid

= 2lw + 2lh + 2wh

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

= 108

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GRE Math : Solid Geometry

Study concepts, example questions & explanations for GRE Math

All GRE Math Resources

Example Questions

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

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

Example Question #3 : How To Find The Diagonal Of A Cube

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

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

Example Question #24 : Solid Geometry

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

Example Question #3 : How To Find The Volume Of A Cube

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

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

All GRE Math Resources

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