GMAT Math : DSQ: Calculating the surface area of a cylinder

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Example Question #203 : Geometry

The city of Wilsonville has a small cylindrical water tank in which it keeps an emergency water supply. Give its surface area, to the nearest hundred square feet.

Statement 1: The water tank holds about 37,700 cubic feet of water.

Statement 2: About ten and three fourths gallons of paint, which gets about 350 square feet of coverage per gallon can, will need to be used to paint the tank completely.

Possible Answers:

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.

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.

Correct answer:

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

Explanation:

Statement 1 is unhelpful in that it gives the volume, not the surface area, of the tank. The volume of a cylinder depends on two independent values, the height and the area of a base; neither can be determined, so neither can the surface area.

From Statement 2 alone, we can find the surface area. One gallon of paint covers 350 square feet, so, since $10 \frac{3}{4}$ gallons of this paint will cover about

$10 \frac{3}{4} \cdot 350 \approx 3,800$ square feet, the surface area of the tank.

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Example Question #2431 : Gmat Quantitative Reasoning

Of a given cylinder and a given sphere, which, if either, has the greater surface area?

Statement 1: The height of the cylinder is equal to the radius of the sphere.

Statement 2: The radius of a base of the cylinder is greater than the radius of the sphere.

Possible Answers:

BOTH statements TOGETHER are insufficient 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.

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.

Correct answer:

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

Explanation:

The surface area of the cylinder can be calculated from radius and height using the formula:

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

Let the radius of the sphere be . Then its surface area can be calculated to be

$A = 4 \pi R^{2}$

From Statement 1 alone, h = R , so the surface area of the cylinder can be expressed as

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

We cannot compare this to the surface area of the sphere without knowing anything about the radius of a base.

Likewise, from Statement 2 alone, we know that r > R , but without knowing anything about the height, we cannot compare the surface areas.

Now assume both statements. Again, from Statement 1,

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

Since, from Statement 2, r > R , if follows that

$2 \pi R \cdot r >2 \pi R \cdot R$ , or, $2 \pi r R > 2 \pi R ^{2}$ .

It also follows that

$r^{2} > R^{2}$

and

$2 \pi r^{2} > 2 \pi R^{2}$

Therefore, we can add both sides of the inequalities:

$2 \pi r R + 2 \pi r^{2} > 2 \pi R^{2}+ 2 \pi R ^{2}$

$2 \pi r R + 2 \pi r^{2} > 4 \pi R^{2}$

and

a > A ,

so the cylinder has the greater surface area of the two solids.

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Example Question #12 : Dsq: Calculating The Surface Area Of A Cylinder

Inscribed_cylinder

In the above figure, a cylinder is inscribed inside a cube. and mark the points of tangency the upper base has with $\overline{BC}$ and $\overline{CD}$ . What is the surface area of the cylinder?

Statement 1: Arc $\widehat{XY}$ has length $5 \pi$ .

Statement 2: Arc $\widehat{XY}$ has degree measure $90 ^{\circ }$ .

Possible Answers:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is 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 to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

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

Explanation:

Assume Statement 1 alone. Since $\widehat{XY}$ has length one fourth the circumference of a base, then each base has circumference $5 \pi \cdot 4 = 20 \pi$ , and radius $r = C \div \left (2\pi \right )= 20 \pi \div \left (2 \pi \right )= 10$ . It follows that each base has area $A = \pi r^{2}= \pi \cdot 10^{2}= 100 \pi$

Also, the diameter is $C \div \pi = 20 \pi \div \pi = 20$ ; it is also the length of each edge, and it is therefore the height. The lateral area is the product of height 20 and circumference $20 \pi$ , or $20 \cdot 20 \pi = 400 \pi$ .

The surface area can now be calculated as the sum of the areas:

$400 \pi + 100 \pi+ 100 \pi= 600 \pi$ .

Statement 2 is actually a redundant statement; since each base is inscribed inside a square, it already follows that $\widehat{XY}$ is one fourth of a circle - that is, a $90^{\circ }$ arc.

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Example Question #14 : Dsq: Calculating The Surface Area Of A Cylinder

Of a given cylinder and a given cube, which, if either, has the greater surface area?

Statement 1: Both the height of the cylinder and the diameter of its bases are equal to the length of one edge of the cube.

Statement 2: Each face of the cube has as its area four times the square of the radius of the bases of the cylinder.

Possible Answers:

EITHER 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.

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.

Correct answer:

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

Explanation:

The surface area of a cylinder, given height and radius of the bases , is given by the formula

$A =2 \pi r (r+h)$

The surface area of a cube, given the length of each edge, is given by the formula

$A ' = 6s^{2}$ .

Assume Statement 1 alone. Then h = s and, since the diameter of a base is , the radius is half this, or $r = \frac{1}{2}s$ . The surface area of the cylinder is, in terms of , equal to

$A =2 \pi s \left ( s+ \frac{1}{2}s \right )= 2 \pi s \cdot \frac{3}{2}s = 3 \pi s^{2}$ .

Since $3 \pi > 6$ , the cylinder has the greater area regardless of the actual measurements.

Assume Statement 2 alone. The cube has six faces with area $4r^{2}$ , so its surface area is six times this, or $6 \cdot 4r^{2} = 24 r^{2}$ . The surface area of the cylinder is $A =2 \pi r (r+h) = 2 \pi r^{2}+ 2 \pi rh$ ; however, without knowing anything aobout the height of the cylinder, we cannot compare the two surface areas.

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Example Question #15 : Dsq: Calculating The Surface Area Of A Cylinder

Of Cylinder 1 and Cylinder 2, which, if either, has the greater surface area?

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

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

Possible Answers:

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.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

The surface area of the cylinder can be calculated from radius and height using the formula:

$A = 2 \pi rh + 2 \pi r^{2}$ .

We show that both statements together provide insufficient information by first noting that if the two cylinders have the same height, and their bases have the same radius, their surface areas will be the same.

Now we explore the case in which Cylinder 1 has height 6 and bases with radius 8, and Cylinder 2 has height 8 and bases of radius 6.

The surface area of Cylinder 1 is

$A = 2 \pi \cdot 8 \cdot 6 + 2 \pi \cdot 8^{2}$

$= 2 \pi \cdot 8 \cdot 6 + 2 \pi \cdot 8 \cdot 8$

$=96 \pi + 128 \pi$

$=224 \pi$

The surface area of Cylinder 2 is

$A = 2 \pi \cdot 6 \cdot 8 + 2 \pi \cdot 6^{2}$

$= 2 \pi \cdot 6 \cdot 8 + 2 \pi \cdot 6 \cdot 6$

$=96 \pi + 72\pi$

$=168\pi$

In this scenario, Cylinder 1 has the greater surface area.

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Example Question #16 : Dsq: Calculating The Surface Area Of A Cylinder

Inscribed_cylinder

In the above figure, a cylinder is inscribed inside a cube. What is the surface area of the cylinder?

Statement 1: The volume of the cube is 729.

Statement 2: The surface area of the cube is 486.

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.

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:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

The surface area of the cylinder can be calculated from radius and height using the formula:

$A = 2 \pi rh + 2 \pi r^{2}$ .

It can be seen from the diagram that if we let be the length of one edge of the cube, then h = s and $r = \frac{1}{2}s$ . The surface area formula can be rewritten as

$A = 2 \pi \cdot \frac{1}{2}s \cdot s + 2 \pi \left ( \frac{1}{2}s \right ) ^{2}$

$A = \pi s ^{2}+ 2 \pi \cdot \frac{1}{4}s ^{2}$

$A = \pi s ^{2}+ \frac{1}{2} \pi s ^{2}$

$A = \frac{3}{2} \pi s ^{2}$

Subsequently, the length of one edge of the cube is sufficient to calculate the surface area of the cylinder.

From Statement 1 alone, the length of an edge of the cube can be calculated using the volume formula:

$s^{3} = V = 729$

$s = \sqrt[3]{729}= 9$

From Statement 2 alone, the length of an edge of the cube can be calculated using the surface area formula:

$6s^{2} = A = 486$

$s^{2}= 486 \div 6 = 81$

$s = \sqrt{81}= 9$

Since can be calculated from either statement alone, so can the surface area of the cylinder:

$A = \frac{3}{2} \pi s ^{2} = \frac{3}{2} \pi \cdot 9 ^{2} = \frac{243}{2} \pi$

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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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GMAT Math : DSQ: Calculating the surface area of a cylinder

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #203 : Geometry

Example Question #2431 : Gmat Quantitative Reasoning

Example Question #12 : Dsq: Calculating The Surface Area Of A Cylinder

Example Question #14 : Dsq: Calculating The Surface Area Of A Cylinder

Example Question #15 : Dsq: Calculating The Surface Area Of A Cylinder

Example Question #16 : Dsq: Calculating The Surface Area Of A Cylinder

Example Question #211 : Geometry

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