GRE Math : Solid Geometry

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Example Questions

Example Question #1 : Cylinders

The area of the base of a circular right cylinder is quadrupled. By what percentage is the outer face increased by this change?

Possible Answers:

200%

400%

250%

300%

100%

Correct answer:

100%

Explanation:

The base of the original cylinder would have been πr², and the outer face would have been 2πrh, where h is the height of the cylinder.

Let's represent the original area with A, the original radius with r, and the new radius with R: therefore, we know πR² = 4A, or πR² = 4πr². Solving for R, we get R = 2r; therefore, the new outer face of the cylinder will have an area of 2πRh or 2π2rh or 4πrh, which is double the original face area; thus the percentage of increase is 100%. (Don't be tricked into thinking it is 200%. That is not the percentage of increase.)

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

What is the surface area of a cylinder with a radius of 17 and a height of 3?

Possible Answers:

2205

2000

3107

1984

2137

Correct answer:

2137

Explanation:

We need the formula for the surface area of a cylinder: SA = 2πr² + 2πrh. This formula has π in it, but the answer choices don't. This means we must approximate π. None of the answers are too close to each other so we could really even use 3 here, but it is safest to use 3.14 as an approximate value of π.

Then SA = 2 * 3.14 * 17² + 2 * 3.14 * 17 * 3 ≈ 2137

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Example Question #2 : Solid Geometry

What is the surface area of a cylinder with a radius of 6 and a height of 9?

Possible Answers:

96π

64π

225π

108π

180π

Correct answer:

180π

Explanation:

surface area of a cylinder

= 2πr² + 2πrh

= 2π * 6² + 2π * 6 *9

= 180π

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Example Question #1 : Solid Geometry

Quantitative Comparison

Quantity A: The volume of a cylinder with a radius of 3 and a height of 4

Quantity B: 3 times the volume of a cone with a radius of 3 and a height of 4

Possible Answers:

Quantity A is greater.

The two quantities are equal.

Quantity B is greater.

The relationship cannot be determined from the information given.

Correct answer:

The two quantities are equal.

Explanation:

There is no need to do the actual computations here to find the two volumes. The volume of a cone is exactly 1/3 the volume of a cylinder with the same height and radius. That means the two quantities are equal. The formulas show this relationship as well: volume of a cone = πr²h/3 and volume of a cylinder = πr²h.

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

A right circular cylinder of volume $200\pi$ has a height of 8.

Quantity A: 10

Quantity B: The circumference of the base

Possible Answers:

Quantity A is greater

The two quantities are equal

The relationship cannot be determined from the information provided.

Quantity B is greater

Correct answer:

Quantity B is greater

Explanation:

The volume of any solid figure is $base \times height$ . In this case, the volume of the cylinder is $200\pi$ and its height is , which means that the area of its base must be $25\pi$ . Working backwards, you can figure out that the radius of a circle of area $25\pi$ is . The circumference of a circle with a radius of is $10\pi$ , which is greater than .

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

What is the surface area of a cylinder that has a diameter of 6 inches and is 4 inches tall?

Possible Answers:

$12\pi$

$4\pi$

$42\pi$

$10\pi$

$15\pi$

Correct answer:

$42\pi$

Explanation:

The formula for the surface area of a cylinder is $\dpi{100} 2\pi r^{2}+ 2\pi rh$ ,

where is the radius and is the height.

$\dpi{100} SA=2\pi (3)^{2}+2\pi (3)(4)$

$\dpi{100} =42\pi$

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

A cylinder has a radius of 4 and a height of 8. What is its surface area?

Possible Answers:

$96\pi$

$100\pi$

$64\pi$

$128\pi$

$32\pi$

Correct answer:

$96\pi$

Explanation:

This problem is simple if we remember the surface area formula!

$SA = 2\pi r^2 +2\pi rh = 2\pi *16+2\pi *4*8 = 32\pi +64\pi = 96\pi$

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Example Question #6 : How To Find The Surface Area Of A Cylinder

Quantitative Comparison

Quantity A: Surface area of a cylinder that is 2 feet high and has a radius of 4 feet

Quantity B: Surface area of a box that is 3 feet wide, 2 feet high, and 4 feet long

Possible Answers:

Quantity A is greater.

Quantity B is greater.

The relationship cannot be determined from the information given.

The two quantities are equal.

Correct answer:

Quantity A is greater.

Explanation:

Quantity A: SA of a cylinder = 2πr² + 2πrh = 2π * 16 + 2π * 4 * 2 = 48π

Quantity B: SA of a rectangular solid = 2ab + 2bc + 2ac = 2 * 3 * 2 + 2 * 2 * 4 + 2 * 3 * 4 = 52

48π is much larger than 52, because π is approximately 3.14.

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

A cylinder has a height of 4 and a circumference of 16π. What is its volume

Possible Answers:

128π

64π

256π

none of these

16π

Correct answer:

256π

Explanation:

circumference = πd

d = 2r

volume of cylinder = πr²h

r = 8, h = 4

volume = 256π

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

Rusty is considering making a cylindrical grain silo to store his crops. He has an area that is 6 feet long, 6 feet wide and 12 feet tall to build a cylinder in. What is the maximum volume of grain that he can store in this cylinder?

Possible Answers:

$54\pi$

$108\pi$

$432\pi$

108

432

Correct answer:

$108\pi$

Explanation:

The maximum cylindrical base can have a diameter of 6 and therefore a radius of 3. The formula for the volume of a cylinder is $\dpi{100} \small \pi r^{2}h$ , which in this case is $\dpi{100} \small 3\times 3\times 12\times \pi$ .

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GRE Math : Solid Geometry

Study concepts, example questions & explanations for GRE Math

All GRE Math Resources

Example Questions

Example Question #1 : Cylinders

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

Example Question #2 : Solid Geometry

Example Question #1 : Solid Geometry

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

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

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

Example Question #6 : How To Find The Surface Area Of A Cylinder

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

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

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