How to find the volume of a cylinder - SAT Math

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Question

A cylinder has a height that is three times as long as its radius. If the lateral surface area of the cylinder is 54π square units, then what is its volume in cubic units?

Answer

Let us call r the radius and h the height of the cylinder. We are told that the height is three times the radius, which we can represent as h = 3r.

We are also told that the lateral surface area is equal to 54π. The lateral surface area is the surface area that does not include the bases. The formula for the lateral surface area is equal to the circumference of the cylinder times its height, or 2πrh. We set this equal to 54π,

2πrh = 54π

Now we substitute 3r in for h.

2πr(3r) = 54π

6πr2 = 54π

Divide by 6π

r2 = 9.

Take the square root.

r = 3.

h = 3r = 3(3) = 9.

Now that we have the radius and the height of the cylinder, we can find its volume, which is given by πr2h.

V = πr2h

V = π(3)2(9) = 81π

The answer is 81π.

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Question

What is the volume of a hollow cylinder whose inner radius is 2 cm and outer radius is 4 cm, with a height of 5 cm?

Answer

The volume is found by subtracting the inner cylinder from the outer cylinder as given by V = πrout2 h – πrin2 h. The area of the cylinder using the outer radius is 80π cm3, and resulting hole is given by the volume from the inner radius, 20π cm3. The difference between the two gives the volume of the resulting hollow cylinder, 60π cm3.

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Question

The volume of a cylinder is 36π. If the cylinder’s height is 4, what is the cylinder’s diameter?

Answer

Volume of a cylinder? V = πr2h. Rewritten as a diameter equation, this is:

V = π(d/2)2h = πd2h/4

Sub in h and V: 36p = πd2(4)/4 so 36p = πd2

Thus d = 6

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Question

Two cylinders are full of milk. The first cylinder is 9” tall and has a base diameter of 3”. The second cylinder is 9” tall and has a base diameter of 4”. If you are going to pour both cylinders of milk into a single cylinder with a base diameter of 6”, how tall must that cylinder be for the milk to fill it to the top?

Answer

Volume of cylinder = π * (base radius)2 x height = π * (base diameter / 2 )2 x height

Volume Cylinder 1 = π * (3 / 2 )2 x 9 = π * (1.5 )2 x 9 = π * 20.25

Volume Cylinder 2 = π * (4 / 2 )2 x 9 = π * (2 )2 x 9 = π * 36

Total Volume = π * 20.25 + π * 36

Volume of Cylinder 3 = π * (6 / 2 )2 x H = π * (3 )2 x H = π * 9 x H

Set Total Volume equal to the Volume of Cylinder 3 and solve for H

π * 20.25 + π * 36 = π * 9 x H

20.25 + 36 = 9 x H

H = (20.25 + 36) / 9 = 6.25”

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Question

A cylinder has a height of 5 inches and a radius of 3 inches. Find the lateral area of the cylinder.

Answer

LA = 2π(r)(h) = 2π(3)(5) = 30π

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Question

A cylinder has a volume of 20. If the radius doubles, what is the new volume?

Answer

The equation for the volume of the cylinder is πr2h. When the radius doubles (r becomes 2r) you get π(2r)2h = 4πr2h. So when the radius doubles, the volume quadruples, giving a new volume of 80.

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Question

A metal cylindrical brick has a height of $6\ in$ . The area of the top is $25\pi\ in^2$ . A circular hole with a radius of $2\ in$ is centered and drilled half-way down the brick. What is the volume of the resulting shape?

Answer

To find the final volume, we will need to subtract the volume of the hole from the total initial volume of the cylinder.

The volume of a cylinder is given by the product of the base area times the height: $V=A_{base}h$ .

Find the initial volume using the given base area and height.

$V_i=(25\pi)(6)=150\pi\ in^3$

Next, find the volume of the hole that was drilled. The base area of this cylinder can be calculated from the radius of the hole. Remember that the height of the hole is only half the height of the block.

$V_{hole}=(\pi r^2)h$

$V_{hole}=(\pi(2)^2)(3)=(4\pi)(3)=12\pi\ in^3$

Finally, subtract the volume of the hole from the total initial volume.

$150\pi-12\pi=138\pi\ in^3$

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Question

What is the volume of a right cylinder with a circumference of 25π in and a height of 41.3 in?

Answer

The formula for the volume of a right cylinder is: V = A * h, where A is the area of the base, or πr2. Therefore, the total formula for the volume of the cylinder is: V = πr2h.

First, we must solve for r by using the formula for a circumference (c = 2πr): 25π = 2πr; r = 12.5.

Based on this, we know that the volume of our cylinder must be: π12.5241.3 = 6453.125π in3

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Question

What is the volume of a cylinder with a diameter of 13 inches and a height of 27.5 inches?

Answer

The equation for the volume of a cylinder is V = Ah, where A is the area of the base and h is the height.

Thus, the volume can also be expressed as V = πr2h.

The diameter is 13 inches, so the radius is 13/2 = 6.5 inches.

Now we can easily calculate the volume:

V = 6.52π * 27.5 = 1161.88π in3

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Question

An 8-inch cube has a cylinder drilled out of it. The cylinder has a radius of 2.5 inches. To the nearest hundredth, approximately what is the remaining volume of the cube?

Answer

We must calculate our two volumes and subtract them. The volume of the cube is very simple: 8 * 8 * 8, or 512 in3.

The volume of the cylinder is calculated by multiplying the area of its base by its height. The height of the cylinder is 8 inches (the height of the cube through which it is being drilled). Therefore, its volume is πr2h = π * 2.52 * 8 = 50π in3

The volume remaining in the cube after the drilling is: 512 – 50π, or approximately 512 – 157.0795 = 354.9205, or 354.92 in3.

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Question

An 12-inch cube of wood has a cylinder drilled out of it. The cylinder has a radius of 3.75 inches. If the density of the wood is 4 g/in3, what is the mass of the remaining wood after the cylinder is drilled out?

Answer

We must calculate our two volumes and subtract them. Following this, we will multiply by the density.

The volume of the cube is very simple: 12 * 12 * 12, or 1728 in3.

The volume of the cylinder is calculated by multiplying the area of its base by its height. The height of the cylinder is 8 inches (the height of the cube through which it is being drilled). Therefore, its volume is πr2h = π * 3.752 * 12 = 168.75π in3.

The volume remaining in the cube after the drilling is: 1728 – 168.75π, or approximately 1728 – 530.1433125 = 1197.8566875 in3. Now, multiply this by 4 to get the mass: (approx.) 4791.43 g.

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Question

A hollow prism has a base 5 in x 6 in and a height of 10 in. A closed, cylindrical can is placed in the prism. The remainder of the prism is then filled with gel around the cylinder. The thickness of the can is negligible. Its diameter is 4 in and its height is half that of the prism. What is the approximate volume of gel needed to fill the prism?

Answer

The general form of our problem is:

Gel volume = Prism volume – Can volume

The prism volume is simple: 5 * 6 * 10 = 300 in3

The volume of the can is found by multiplying the area of the circular base by the height of the can. The height is half the prism height, or 10/2 = 5 in. The area of the base is equal to πr_2. Note that the prompt has given the diameter. Therefore, the radius is 2, not 4. The base's area is: 22_π = 4_π_. The total volume is therefore: 4_π_ * 5 = 20_π_ in3.

The gel volume is therefore: 300 – 20_π_ or (approx.) 237.17 in3.

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Question

A hollow prism has a base 12 in x 13 in and a height of 42 in. A closed, cylindrical can is placed in the prism. The remainder of the prism is then filled with gel, surrounding the can. The thickness of the can is negligible. Its diameter is 9 in and its height is one-fourth that of the prism. The can has a mass of 1.5 g per in3, and the gel has a mass of 2.2 g per in3. What is the approximate overall mass of the contents of the prism?

Answer

We must find both the can volume and the gel volume. The formula for the gel volume is:

Gel volume = Prism volume – Can volume

The prism volume is simple: 12 * 13 * 42 = 6552 in3

The volume of the can is found by multiplying the area of the circular base by the height of the can. The height is one-fourth the prism height, or 42/4 = 10.5 in. The area of the base is equal to πr_2. Note that the prompt has given the diameter. Therefore, the radius is 4.5, not 9. The base's area is: 4.52_π = 20.25_π_. The total volume is therefore: 20.25_π_ * 10.5 = 212.625_π_ in3.

The gel volume is therefore: 6552 – 212.625_π_ or (approx.) 5884.02 in3.

The approximate volume for the can is: 667.98 in3

From this, we can calculate the approximate mass of the contents:

Gel Mass = Gel Volume * 2.2 = 5884.02 * 2.2 = 12944.844 g

Can Mass = Can Volume * 1.5 = 667.98 * 1.5 = 1001.97 g

The total mass is therefore 12944.844 + 1001.97 = 13946.814 g, or approximately 13.95 kg.

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Question

Jessica wishes to fill up a cylinder with water at a rate of gallons per minute. The volume of the cylinder is gallons. The hole at the bottom of the cylinder leaks out gallons per minute. If there are gallons in the cylinder when Jessica starts filling it, how long does it take to fill?

Answer

Jessica needs to fill up gallons at the effective rate of $1.5:\text{gal/min}$ . $12: \text{gal}$ divided by $1.5: \text{gal/min}$ is equal to $8 :\text{min}$ . Notice how the units work out.

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Question

A vase needs to be filled with water. If the vase is a cylinder that is $\dpi{100} \small 12{}''$ tall with a $\dpi{100} \small 2{}''$ radius, how much water is needed to fill the vase?

Answer

Cylinder

$\dpi{100} \small V = \pi r^{2}h$

$\dpi{100} \small V = \pi (2)^{2}\times 12$

$\dpi{100} \small V = 4\times 12\pi$

$\dpi{100} \small V = 48\pi$

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Question

A cylinder has a base diameter of 12 in and is 2 in tall. What is the volume?

Answer

The volume of a cylinder is $V=\pi r^{2}h$

The diameter is given, so make sure to divide it in half.

The units are inches cubed in this example

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Question

A circle has a circumference of $4\pi$ and it is used as the base of a cylinder. The cylinder has a surface area of $16\pi$ . Find the volume of the cylinder.

Answer

Using the circumference, we can find the radius of the circle. The equation for the circumference is $2\pi r$ ; therefore, the radius is 2.

Now we can find the area of the circle using $\pi r^{2}$ . The area is $4\pi$ .

Finally, the surface area consists of the area of two circles and the area of the mid-section of the cylinder: $2\cdot 4\pi +4\pi h=16\pi$ , where $h$ is the height of the cylinder.

Thus, $h=2$ and the volume of the cylinder is $4\pi h=4\pi \cdot 2=8\pi$ .

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Question

What is the volume of a cylinder with a radius of 4 and a height of 5?

Answer

$volume = \pi r^{2}h = \pi \cdot 4^{2} \cdot 5 = 80\pi$

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Question

Claire's cylindrical water bottle is 9 inches tall and has a diameter of 6 inches. How many cubic inches of water will her bottle hold?

Answer

The volume is the area of the base times the height. The area of the base is $\pi r^{2}$ , and the radius here is 3. $\pi\cdot 3^{2}\cdot 9=81\pi$

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Question

What is the volume of a circular cylinder whose height is 8 cm and has a diameter of 4 cm?

Answer

The volume of a circular cylinder is given by $V = \pi r^{2}h$ where is the radius and is the height. The diameter is given as 4 cm, so the radius would be 2 cm as the diameter is twice the radius.

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