Cylinders - GMAT Quantitative

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Question

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

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Answer

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

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

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Question

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?

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Answer

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

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

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

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

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Question

Calculate the surface area of the following cylinder.

(Not drawn to scale.)

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Answer

The equation for the surface area of a cylinder is:

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

we plug in our values: to find the surface area

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

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

$SA=88\pi$

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Question

Calculate the surface area of the following cylinder.

(Not drawn to scale.)

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Answer

The equation for the surface area of a cylinder is

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

We plug in our values 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$$

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

$SA=130\pi$

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Question

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

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Answer

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$

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

$SA=32\pi+72\pi$

$SA=104\pi$

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Question

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

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Answer

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

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

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

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

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

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

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Question

Find the surface area of a cylinder whose height is and radius is .

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Answer

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

Thus,

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Question

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

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Answer

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

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

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

$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}$$

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

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Question

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

Tap to reveal answer

Answer

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

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

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Question

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?

Tap to reveal answer

Answer

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

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

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

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

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Question

Calculate the surface area of the following cylinder.

(Not drawn to scale.)

Tap to reveal answer

Answer

The equation for the surface area of a cylinder is:

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

we plug in our values: to find the surface area

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

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

$SA=88\pi$

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Question

Calculate the surface area of the following cylinder.

(Not drawn to scale.)

Tap to reveal answer

Answer

The equation for the surface area of a cylinder is

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

We plug in our values 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$$

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

$SA=130\pi$

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Question

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

Tap to reveal answer

Answer

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$

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

$SA=32\pi+72\pi$

$SA=104\pi$

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Question

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

Tap to reveal answer

Answer

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

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

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

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

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

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

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Question

Find the surface area of a cylinder whose height is and radius is .

Tap to reveal answer

Answer

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

Thus,

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Question

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

Tap to reveal answer

Answer

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

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

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

$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}$$

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

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Question

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

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Answer

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}$$h}{3} = $\frac{picdot $6^{2}$$cdot 7}{3} = 84pi

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Question

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

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Answer

volume = $\frac{4}{3}$pi $r^{3}$ = $\frac{4}{3}$pitimes $9^{3}$ = 972pi

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Question

What is the volume of a cylinder that is 12 inches high and has a radius of 6 inches?

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Answer

$volume=\pi $r^{2}$h=\pi $6^{2}$12=432\pi$

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Question

A cylindrical gas tank is 30 meters high and has a radius of 10 meters. How much oil can the tank hold?

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Answer

$V=\pi $r^{2}$h=\pi $(10^{2}$)30=3000\pi$

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