### All GRE Math Resources

## Example Questions

### Example Question #1 : Solid Geometry

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:**

100%

200%

400%

250%

300%

**Correct answer:**

100%

The base of the original cylinder would have been πr^{2}, 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 πR^{2} = 4A, or πR^{2} = 4πr^{2}. 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*.)

### 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:**

2137

2205

2000

3107

1984

**Correct answer:**

2137

We need the formula for the surface area of a cylinder: SA = 2*πr*^{2} + 2*πrh*. 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 * 17^{2} + 2 * 3.14 * 17 * 3 ≈ 2137

### 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:**

225*π*

180*π*

108*π*

96π

64*π*

**Correct answer:**

180*π*

surface area of a cylinder

= 2*πr*^{2} + 2*πrh*

= 2*π ** 6^{2} + 2*π ** 6 *9

= 180*π*

### Example Question #2 : 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 B is greater.

The relationship cannot be determined from the information given.

The two quantities are equal.

Quantity A is greater.

**Correct answer:**

The two quantities are equal.

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 = *πr*^{2}*h*/3 and volume of a cylinder = *πr*^{2}*h*.

### Example Question #2 : Solid Geometry

A right circular cylinder of volume has a height of 8.

Quantity A: 10

Quantity B: The circumference of the base

**Possible Answers:**

Quantity B is greater

The relationship cannot be determined from the information provided.

The two quantities are equal

Quantity A is greater

**Correct answer:**

Quantity B is greater

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

### Example Question #3 : Solid Geometry

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

**Possible Answers:**

**Correct answer:**

The formula for the surface area of a cylinder is ,

where is the radius and is the height.

### Example Question #4 : Solid Geometry

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

**Possible Answers:**

**Correct answer:**

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

### Example Question #5 : Solid Geometry

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:**

The relationship cannot be determined from the information given.

Quantity A is greater.

Quantity B is greater.

The two quantities are equal.

**Correct answer:**

Quantity A is greater.

Quantity A: SA of a cylinder = 2*πr*^{2} + 2*πrh* = 2*π * *16 + 2*π* * 4 * 2 = 48*π*

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

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

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