GMAT Math : Geometry

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Example Questions

Example Question #211 : Geometry

Of two given solids - a cylinder and a regular triangular prism - which has the greater surface area?

Statement 1: Each side of a triangular base of the prism has length one third the circumference of a base of the cylinder.

Statement 2: The cylinder and the prism have the same height.

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

Statement 1 alone provides insufficient information. The surface area of a cylinder with bases of radius and height is

$a = 2 \pi r ^{2}+ 2 \pi r h$ .

The surface area of a regular triangular prism with height and whose (equilateral triangular) bases have common sidelength - and perimeter - is

$A = 2 \cdot \frac{S^{2}\sqrt{3}}{4} + 3S \cdot H$ , or

$A = \frac{S^{2}\sqrt{3}}{2} + 3S H$ .

Statement 1 alone tells us that $S = \frac{1}{3} \cdot 2 \pi r$ , or $S = \frac{2 \pi }{3} r$ . However, without any information about the heights, we cannot compare and . Similarly, Statement 2 alone tells us that h = H , but nothing about or .

However, if we combine what we know from Statement 2 with the information from Statement 1, we can answer the question. The surface area of a cylinder or a prism is equal to its lateral area plus the areas of its two congruent bases.

The lateral area of a cylinder or a prism is the height multiplied by the perimeter or circumference of a base. From Statement 1, the circumference of the bases of the cylinder is equal to the perimeter of the bases of the prism, and from Statement 2, the heights are equal. It follows that the lateral areas are equal, and that the figure with the bases that are greater in area has the greater surface area.

For simplicity's sake, we will assume that the circumference of the base of the cylinder is $6 \pi$ , for reasons that will be apparent later; this reasoning works for any circumference. The radius of this base is $6 \pi \div 2 \pi = 3$ , and the area is $\pi \cdot 3^{2} = 9 \pi$ . The perimeter of the triangular base of the prism is also $6 \pi$ , so its sidelength is $2 \pi$ , making its area $\frac{(2 \pi)^{2}\sqrt{3}}{2} = 2 \pi^{2}\sqrt{3}$ , which is greater than $9 \pi$ . This makes the prism the greater in surface area as well.

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Example Question #41 : Rectangular Solids & Cylinders

What is the volume of the cylinder?

Statement 1: the cylinder has a radius of 3

Statement 2: the cylinder has a height of 4

Possible Answers:

Statements 1 and 2 TOGETHER are NOT sufficient.

Statement 1 ALONE is sufficient, but statement 2 is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statement 2 ALONE is sufficient, but statement 1 is not sufficient.

Correct answer:

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Explanation:

The formula for the volume of a cylinder is: $volume = \Pi \ast r^{2}\ast h$

Therefore we need both Statement 1 and 2 to find the volume, so both statements together are sufficient, but neither statement alone is sufficient.

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Example Question #211 : Geometry

Give the radius of a cylinder with volume 1,000 cubic inches.

1) Its height is 40 inches.

2) The area of its base is 25 square inches.

Possible Answers:

BOTH statements TOGETHER are NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is not sufficient.

EITHER Statement 1 or Statement 2 ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is not sufficient.

Correct answer:

EITHER Statement 1 or Statement 2 ALONE is sufficient to answer the question.

Explanation:

The two statements are actually equivalent; if $\small V$ is its volume, $\small B$ is the area of its base, and $\small h$ is its height, then $\small V= B h$ , or $\small B=\frac{V}{h}$ . So if we know the first statement, that is, $\small h=40$ , then $\small B=\frac{1,000}{40} = 25$ , which is the second statement.

To find the radius, use $\small B = \pi r^{2}$ , or, equivalently, $\small r =\sqrt{\frac{B}{\pi }}$

$\small \small \small r =\sqrt{\frac{B}{\pi }}=\sqrt{\frac{25}{\pi }} \approx 2.8$

The answer is that either statement alone is sufficient to answer the question.

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Example Question #212 : Geometry

A factory makes two sizes of cylindrical barrels. How many small cylindrical barrels full of liquid would it take to fill a large barrel?

Statement 1: A large cylindrical barrel is twice as high as a small cylindrical barrel.

Statement 2: A large cylindrical barrel is three times as wide as a small cylindrical barrel.

Possible Answers:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

The question is essentially this: if $V_{S} , V_{L}$ are the volumes of the barrels, then find such that

$V_{L} = KV_{S}$ ,

$\frac{V_{L}}{V_{S}} = K$

Equivalently,

$\frac{ \pi r_{L}^{2} h_{L} }{ \pi r_{S}^{2} h_{S} } = K$

$K = \left ( \frac{ r_{L} } {r_{S} }\right ) ^{2} \cdot \frac{ h_{L} } {h_{S} }$

This means that knowing both the ratio of the heights and the ratio of the radii (and subsequently, the widths) is necessary and sufficient. Therefore, you need both statements, and both together are sufficient.

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Example Question #44 : Rectangular Solids & Cylinders

Which of Cylinder 1 and Cylinder 2, if either, has the greater volume?

Statement 1: The height of Cylinder 1 is equal to the radius of the base of Cylinder 2.

Statement 2: The height of Cylinder 2 is equal to twice the radius of the base of Cylinder 1.

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Assume both statements. We show that insufficient information is provided using two cases.

Case 1: Cylinder 1 has height 10 and radius 20. Then Cylinder 2 has height 40 and radius 10.

The volume of Cylinder 1 is $V = \pi r ^{2} h = \pi \cdot 20 ^{2} \cdot 10 = 8,000\pi$

The volume of Cylinder 2 is $V = \pi r ^{2} h = \pi \cdot 10 ^{2} \cdot 40 = 4,000\pi$

Case 2: Cylinder has height 40 and radius 10. Then Cylinder 2 has height 20 and radius 40.

The volume of Cylinder 1 is $V = \pi r ^{2} h = \pi \cdot 10 ^{2} \cdot 40 = 4,000\pi$

The volume of Cylinder 2 is $V = \pi r ^{2} h = \pi \cdot 40 ^{2} \cdot 20 = 32,000\pi$

In one case, Cylinder 1 had greater volume; in the other, Cylinder 2 did. This makes the two statements insufficient.

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Example Question #213 : Geometry

How many identical cans can be packed in a certain box?

(1) The box is centimeters wide and centimeters high.

(2) Each can is centimeters high.

Possible Answers:

C: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient

E: Statements (1) and (2) TOGETHER are not sufficient

D: EACH statement ALONE is sufficient

A: Statement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient

B: Statement (2) ALONE is sufficient, but statement (1) ALONE is not sufficient

Correct answer:

E: Statements (1) and (2) TOGETHER are not sufficient

Explanation:

(1) does not give us any information about the cans' dimension or the length of the box, so it is not sufficient alone.

(2) does not give us box dimensions or any information about the cans' diameter, so it is not sufficient alone.

Both statement taken together give more information but we still dont have complete dimensions for the box or each can.

Therefore the right answer is E.

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Example Question #46 : Rectangular Solids & Cylinders

Jenkins has a poster tube which he is using to carry his posters to college.

I) The poster tube has a volume of 46 in^3 .

II) The poster tube is inches long.

What is the radius of the poster tube?

Possible Answers:

Either statement alone is sufficient to answer the question.

Neither I nor II are sufficient to answer the question. More information is needed.

Statement II is sufficient to answer the question, but statement I is not sufficient to answer the question.

Both statements are necessary to answer the question.

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

Correct answer:

Both statements are necessary to answer the question.

Explanation:

Recall the formula for volume of a cylinder:

$\small V=\pi r^2h$

Statement I gives us V

Statement II gives us h

We can then use both of them to find our "r." We cannot do it without both of them.

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Example Question #24 : Cylinders

How much water, in cubic feet, can a cylindrical water tank whose bases have radius 6 feet hold?

Statement 1: The lateral area of the tank is 125.66 square yards.

Statement 2: The tank is 30 feet high.

Possible Answers:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is not sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is not sufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

We are given the radius; if we know the height, we can use the formula

$V=\pi r^{2}h$

to calculate the volume of the tank.

The second statement gives us that the tank is 30 feet high. But the first statement gives us the way to find the height by using the lateral area formula.

First we have to convert square yards to square feet by multiplying by 9.

$125.66 \cdot 9 = 1,130.94$

$L.A.= 2\pi rh$

$1,130.94= 2\pi \cdot 6\cdot h= 12\pi h$

$h=\frac{ 1,130.94}{12\pi }=30$

Either way, we now have both radius and height, and we can find the volume:

$V=\pi \cdot 6^{2} \cdot 30 \approx 3392.9$

The answer is that either statement alone is sufficient to answer the question.

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Example Question #217 : Geometry

What is the length of the edge of a cube?

1) Its volume is 1,728 cubic meters.

2) Its surface area is 864 square meters

Possible Answers:

EACH statement ALONE is sufficient.

Statements 1 and 2 TOGETHER are not sufficient.

Statement 2 ALONE is sufficient, but Statement 1 alone is not sufficient.

BOTH statements TOGETHER are sufficient, but neither statement ALONE is sufficient.

Statement 1 ALONE is sufficient, but Statement 2 alone is not sufficient.

Correct answer:

EACH statement ALONE is sufficient.

Explanation:

Call the sidelength, surface area, and volume of the cube , , and , respectively.

Then

$V = s^{3}$

or, equivalently,

$s = \sqrt[3]{V}$

So, given statement 1 alone - that is, given only the volume, you can demonstrate the sidelength to be

$s = \sqrt[3]{1,728} = 12$

Also,

$A = 6s^{2}$

or, equivalently,

$s = \sqrt{\frac{A}{6}}$

Given statement 2 alone - that is, given only the surface area, you can demonstrate the sidelength to be

$s = \sqrt{\frac{864}{6}} = \sqrt{144} = 12$

Therefore, the answer is that either statement alone is sufficient.

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Example Question #1 : Dsq: Calculating The Length Of An Edge Of A Cube

A sphere is inscribed inside a cube. What is the volume of the sphere?

Statement 1: The surface area of the cube is 216.

Statement 2: The volume of the cube is 216.

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

The diameter of a sphere inscribed inside a cube is equal to the length of one of the edges of a cube. From either the surface area or the volume of a cube, the appropriate formula can be used to calculate this length. Half this is the radius, from which the formula $V = \frac{4}{3} \pi r^{3}$ can be used to find the volume of the sphere.

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GMAT Math : Geometry

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #211 : Geometry

Example Question #41 : Rectangular Solids & Cylinders

Example Question #211 : Geometry

Example Question #212 : Geometry

Example Question #44 : Rectangular Solids & Cylinders

Example Question #213 : Geometry

Example Question #46 : Rectangular Solids & Cylinders

Example Question #24 : Cylinders

Example Question #217 : Geometry

Example Question #1 : Dsq: Calculating The Length Of An Edge Of A Cube

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