GMAT Math : Cylinders

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Example Questions

Example Question #1 : Calculating The Surface Area Of A Cylinder

What is the surface area of a cylinder with a radius of 7 and a height of 3?

Possible Answers:

$\dpi{100} \small 98\pi$ $\displaystyle \dpi{100} \small 98\pi$

$\dpi{100} \small 80\pi$ $\displaystyle \dpi{100} \small 80\pi$

$\dpi{100} \small 120\pi$ $\displaystyle \dpi{100} \small 120\pi$

$\dpi{100} \small 42\pi$ $\displaystyle \dpi{100} \small 42\pi$

$\dpi{100} \small 140\pi$ $\displaystyle \dpi{100} \small 140\pi$

Correct answer:

$\dpi{100} \small 140\pi$ $\displaystyle \dpi{100} \small 140\pi$

Explanation:

All we really need here is to remember the formula for the surface area of a cylinder.

$\dpi{100} \small SA=2\pi r^{2}+2\pi rh=2\pi \left ( 49 \right )+2\pi \left ( 7 \right )\left ( 3 \right )=98\pi +42\pi=140\pi$ $\displaystyle \dpi{100} \small SA=2\pi r^{2}+2\pi rh=2\pi \left ( 49 \right )+2\pi \left ( 7 \right )\left ( 3 \right )=98\pi +42\pi=140\pi$

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Example Question #1 : Calculating The Surface Area Of A Cylinder

The height of a cylinder is twice the circumference of its base. The radius of the base is 9 inches. What is the surface area of the cylinder?

Possible Answers:

$810 \pi \textrm{ in}^{2}$ $\displaystyle 810 \pi \textrm{ in}^{2}$

$\left ( 648 \pi +162 \pi^{2} \right )\textrm{ in}^{2}$ $\displaystyle \left ( 648 \pi +162 \pi^{2} \right )\textrm{ in}^{2}$

$\left ( 162 \pi + 648 \pi^{2} \right )\textrm{ in}^{2}$ $\displaystyle \left ( 162 \pi + 648 \pi^{2} \right )\textrm{ in}^{2}$

$1,620 \pi^{2} \textrm{ in}^{2}$ $\displaystyle 1,620 \pi^{2} \textrm{ in}^{2}$

$810 \pi^{2} \textrm{ in}^{2}$ $\displaystyle 810 \pi^{2} \textrm{ in}^{2}$

Correct answer:

$\left ( 162 \pi + 648 \pi^{2} \right )\textrm{ in}^{2}$ $\displaystyle \left ( 162 \pi + 648 \pi^{2} \right )\textrm{ in}^{2}$

Explanation:

The radius of the base is 9 inches, so its circumference is $2 \pi$ $\displaystyle 2 \pi$ times this, or $2 \pi \cdot 9 = 18 \pi$ $\displaystyle 2 \pi \cdot 9 = 18 \pi$ inches. The height is twice this, or $36 \pi$ $\displaystyle 36 \pi$ inches.

Substitute $r = 9 , h = 36 \pi$ $\displaystyle r = 9 , h = 36 \pi$ in the formula for the surface area of the cylinder:

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

$A = 2 \pi \cdot 9 \cdot (9+36\pi ) = 162 \pi + 648 \pi^{2}$ $\displaystyle A = 2 \pi \cdot 9 \cdot (9+36\pi ) = 162 \pi + 648 \pi^{2}$ square inches

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Example Question #3 : Calculating The Surface Area Of A Cylinder

Calculate the surface area of the following cylinder.

(Not drawn to scale.)

Possible Answers:

$56\pi$ $\displaystyle 56\pi$

$154\pi$ $\displaystyle 154\pi$

$32\pi$ $\displaystyle 32\pi$

$88\pi$ $\displaystyle 88\pi$

Correct answer:

$88\pi$ $\displaystyle 88\pi$

Explanation:

The equation for the surface area of a cylinder is:

$SA=2\pi rh+2\pi r^2$ $\displaystyle SA=2\pi rh+2\pi r^2$

we plug in our values: $\displaystyle r=4, h=7$ to find the surface area

$SA=2\pi (4)(7)+2\pi (4)^2$ $\displaystyle SA=2\pi (4)(7)+2\pi (4)^2$

$SA=2 \pi (28)+32\pi$ $\displaystyle SA=2 \pi (28)+32\pi$

$SA=88\pi$ $\displaystyle SA=88\pi$

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Example Question #2 : Calculating The Surface Area Of A Cylinder

Calculate the surface area of the following cylinder.

(Not drawn to scale.)

Possible Answers:

$360\pi$ $\displaystyle 360\pi$

$288\pi$ $\displaystyle 288\pi$

$130\pi$ $\displaystyle 130\pi$

$208\pi$ $\displaystyle 208\pi$

Correct answer:

$130\pi$ $\displaystyle 130\pi$

Explanation:

The equation for the surface area of a cylinder is

$SA=2\pi rh+2\pi r^2$ $\displaystyle SA=2\pi rh+2\pi r^2$

We plug in our values $\displaystyle r=5, h=8$ into the equation to find our answer.

Note: we were given the diameter of the cylinder (10), in order to find the radius we had to divide the diameter by two.

$SA=2\pi (5)(8)+2\pi (5)^2$ $\displaystyle SA=2\pi (5)(8)+2\pi (5)^2$

$SA=2\pi (40)+2\pi (25)$ $\displaystyle SA=2\pi (40)+2\pi (25)$

$SA=130\pi$ $\displaystyle SA=130\pi$

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Example Question #5 : Calculating The Surface Area Of A Cylinder

A cylinder has a height of 9 and a radius of 4. What is the total surface area of the cylinder?

Possible Answers:

$104\pi$ $\displaystyle 104\pi$

$88\pi$ $\displaystyle 88\pi$

$132\pi$ $\displaystyle 132\pi$

$68\pi$ $\displaystyle 68\pi$

$52\pi$ $\displaystyle 52\pi$

Correct answer:

$104\pi$ $\displaystyle 104\pi$

Explanation:

We are given the height and the radius of the cylinder, which is all we need to calculate its surface area. The total surface area will be the area of the two circles on the bottom and top of the cylinder, added to the surface area of the shaft. If we imagine unfolding the shaft of the cylinder, we can see we will have a rectangle whose height is the same as that of the cylinder and whose width is the circumference of the cylinder. This means our formula for the total surface area of the cylinder will be the following:

$SA=2(\pi r^2)+2\pi rh$ $\displaystyle SA=2(\pi r^2)+2\pi rh$

$SA=2\pi (4)^2+2\pi (4)(9)$ $\displaystyle SA=2\pi (4)^2+2\pi (4)(9)$

$SA=32\pi+72\pi$ $\displaystyle SA=32\pi+72\pi$

$SA=104\pi$ $\displaystyle SA=104\pi$

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Example Question #506 : Geometry

Grant is making a canister out of sheet metal. The canister will be a right cylinder with a height of 150 $\displaystyle 150$ mm. The base of the cylinder will have a radius of $\frac{5}{\pi}$ $\displaystyle \frac{5}{\pi}$ mm. If the canister will have an open top, how many square millimeters of metal does Grant need?

Possible Answers:

$1525\text{ } mm^2$ $\displaystyle 1525\text{ } mm^2$

$375+\frac{25}{\pi}\text{ } mm^2$ $\displaystyle 375+\frac{25}{\pi}\text{ } mm^2$

$1500+\frac{25}{\pi} \text{ } mm^2$ $\displaystyle 1500+\frac{25}{\pi} \text{ } mm^2$

$30+\frac{25}{\pi}\text{ } mm^2$ $\displaystyle 30+\frac{25}{\pi}\text{ } mm^2$

$1500+\frac{25}{\pi}\text{ } mm^2$ $\displaystyle 1500+\frac{25}{\pi}\text{ } mm^2$

Correct answer:

$1500+\frac{25}{\pi} \text{ } mm^2$ $\displaystyle 1500+\frac{25}{\pi} \text{ } mm^2$

Explanation:

This question is looking for the surface area of a cylinder with only 1 base. Our surface area of a cylinder is given by:

$\displaystyle SA=2\pi rh+2\pi r^2$

However, because we only need 1 base, we can change it to:

$\displaystyle SA=2\pi rh+\pi r^2$

We know our radius and height, so simply plug them in and simplify.

$\displaystyle SA=2\pi \left( \frac{5}{\pi}\right)150+\pi\left(\frac{5}{\pi}\right)^2=1500+\frac{25}{\pi}$

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Example Question #1 : Cylinders

Find the surface area of a cylinder whose height is $\displaystyle 6$ and radius is $\displaystyle 3$.

Possible Answers:

$54\pi$ $\displaystyle 54\pi$

$45\pi$ $\displaystyle 45\pi$

$27\pi$ $\displaystyle 27\pi$

$18\pi$ $\displaystyle 18\pi$

Correct answer:

$54\pi$ $\displaystyle 54\pi$

Explanation:

To find the surface area of a cylinder, you must use the following equation.

$SA=2\pi{r}h+2\pi{r^2}$ $\displaystyle SA=2\pi{r}h+2\pi{r^2}$

Thus,

$SA=2\pi(3)(6)+2\pi(3^2)=54\pi$ $\displaystyle SA=2\pi(3)(6)+2\pi(3^2)=54\pi$

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Example Question #2 : Cylinders

A right circular cylinder has bases of radius $\pi ^{x}$ $\displaystyle \pi ^{x}$; its height is $\pi ^{y}$ $\displaystyle \pi ^{y}$. Give its surface area.

Possible Answers:

$\pi^{2 x+y+1}$ $\displaystyle \pi^{2 x+y+1}$

$\pi^{ x+2y+1}$ $\displaystyle \pi^{ x+2y+1}$

$2 \pi^{ x+y+1}$ $\displaystyle 2 \pi^{ x+y+1}$

$\pi ^{ 2x+1 } + \pi ^{x+y+1}$ $\displaystyle \pi ^{ 2x+1 } + \pi ^{x+y+1}$

$\pi ^{ 2y+1 } + \pi ^{x+y+1}$ $\displaystyle \pi ^{ 2y+1 } + \pi ^{x+y+1}$

Correct answer:

$\pi ^{ 2x+1 } + \pi ^{x+y+1}$ $\displaystyle \pi ^{ 2x+1 } + \pi ^{x+y+1}$

Explanation:

The surface area of a cylinder can be calculated from its radius and height as follows:

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

Setting $r = \pi^{x}$ $\displaystyle r = \pi^{x}$ and $h = \pi ^{x}$ $\displaystyle h = \pi ^{x}$:

$A = \pi \cdot ( \pi^{x}) ^{2} + \pi \cdot \pi^{x} \cdot \pi^{y}$ $\displaystyle A = \pi \cdot ( \pi^{x}) ^{2} + \pi \cdot \pi^{x} \cdot \pi^{y}$

$A = \pi ^{1 } \cdot \pi^{2x} + \pi ^{1 } \cdot \pi^{x} \cdot \pi^{y}$ $\displaystyle A = \pi ^{1 } \cdot \pi^{2x} + \pi ^{1 } \cdot \pi^{x} \cdot \pi^{y}$

$A = \pi ^{1 +2x} + \pi ^{1+x+y}$ $\displaystyle A = \pi ^{1 +2x} + \pi ^{1+x+y}$ or $A = \pi ^{ 2x+1 } + \pi ^{x+y+1}$ $\displaystyle A = \pi ^{ 2x+1 } + \pi ^{x+y+1}$

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Example Question #1 : Calculating The Volume Of A Cylinder

What is the volume of a cone with a radius of 6 and a height of 7?

Possible Answers:

$\dpi{100} \small 84\pi$ $\displaystyle \dpi{100} \small 84\pi$

$\dpi{100} \small 96\pi$ $\displaystyle \dpi{100} \small 96\pi$

$\dpi{100} \small 42\pi$ $\displaystyle \dpi{100} \small 42\pi$

$\dpi{100} \small 49\pi$ $\displaystyle \dpi{100} \small 49\pi$

$\dpi{100} \small 36\pi$ $\displaystyle \dpi{100} \small 36\pi$

Correct answer:

$\dpi{100} \small 84\pi$ $\displaystyle \dpi{100} \small 84\pi$

Explanation:

The only tricky part here is remembering the formula for the volume of a cone. If you don't remember the formula for the volume of a cone, you can derive it from the volume of a cylinder. The volume of a cone is simply 1/3 the volume of the cylinder. Then,

$volume = \frac{\pi r^{2}h}{3} = \frac{\pi\cdot 6^{2}\cdot 7}{3} = 84\pi$ $\displaystyle volume = \frac{\pi r^{2}h}{3} = \frac{\pi\cdot 6^{2}\cdot 7}{3} = 84\pi$

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Example Question #1 : Calculating The Volume Of A Cylinder

What is the volume of a sphere with a radius of 9?

Possible Answers:

$\dpi{100} \small 972\pi$ $\displaystyle \dpi{100} \small 972\pi$

$\dpi{100} \small 900\pi$ $\displaystyle \dpi{100} \small 900\pi$

$\dpi{100} \small 243\pi$ $\displaystyle \dpi{100} \small 243\pi$

$\dpi{100} \small 81\pi$ $\displaystyle \dpi{100} \small 81\pi$

$\dpi{100} \small 300\pi$ $\displaystyle \dpi{100} \small 300\pi$

Correct answer:

$\dpi{100} \small 972\pi$ $\displaystyle \dpi{100} \small 972\pi$

Explanation:

$\dpi{100} \small volume = \frac{4}{3}\pi r^{3} = \frac{4}{3}\pi\times 9^{3} = 972\pi$ $\displaystyle \dpi{100} \small volume = \frac{4}{3}\pi r^{3} = \frac{4}{3}\pi\times 9^{3} = 972\pi$

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GMAT Math : Cylinders

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #1 : Calculating The Surface Area Of A Cylinder

Example Question #1 : Calculating The Surface Area Of A Cylinder

Example Question #3 : Calculating The Surface Area Of A Cylinder

Example Question #2 : Calculating The Surface Area Of A Cylinder

Example Question #5 : Calculating The Surface Area Of A Cylinder

Example Question #506 : Geometry

Example Question #1 : Cylinders

Example Question #2 : Cylinders

Example Question #1 : Calculating The Volume Of A Cylinder

Example Question #1 : Calculating The Volume Of A Cylinder

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