How to find the volume of a cylinder - PSAT Math
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A cylinder has a volume of 20. If the radius doubles, what is the new volume?
A cylinder has a volume of 20. If the radius doubles, what is the new volume?
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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.
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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The volume of a cylinder is 36π. If the cylinder’s height is 4, what is the cylinder’s diameter?
The volume of a cylinder is 36π. If the cylinder’s height is 4, what is the cylinder’s diameter?
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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
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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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?
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?
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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”
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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A cylinder has a height of 5 inches and a radius of 3 inches. Find the lateral area of the cylinder.
A cylinder has a height of 5 inches and a radius of 3 inches. Find the lateral area of the cylinder.
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LA = 2π(r)(h) = 2π(3)(5) = 30π
LA = 2π(r)(h) = 2π(3)(5) = 30π
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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?
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?
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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.
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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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?
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?
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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.
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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A metal cylindrical brick has a height of 
. The area of the top is 
. A circular hole with a radius of 
is centered and drilled half-way down the brick. What is the volume of the resulting shape?
A metal cylindrical brick has a height of
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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: 
.
Find the initial volume using the given base area and height.
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.
Finally, subtract the volume of the hole from the total initial volume.
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:
Find the initial volume using the given base area and height.
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.
Finally, subtract the volume of the hole from the total initial volume.
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What is the volume of a right cylinder with a circumference of 25π in and a height of 41.3 in?
What is the volume of a right cylinder with a circumference of 25π in and a height of 41.3 in?
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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: π*12.52*41.3 = 6453.125π in3
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: π*12.52*41.3 = 6453.125π in3
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What is the volume of a cylinder with a diameter of 13 inches and a height of 27.5 inches?
What is the volume of a cylinder with a diameter of 13 inches and a height of 27.5 inches?
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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
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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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?
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?
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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.
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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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?
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?
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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.
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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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?
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?
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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.
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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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?
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?
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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.
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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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?
Jessica wishes to fill up a cylinder with water at a rate of
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Jessica needs to fill up 
gallons at the effective rate of 
. 
divided by 
is equal to 
. Notice how the units work out.
Jessica needs to fill up
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What is the volume of a circular cylinder whose height is 8 cm and has a diameter of 4 cm?
What is the volume of a circular cylinder whose height is 8 cm and has a diameter of 4 cm?
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The volume of a circular cylinder is given by 
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.
The volume of a circular cylinder is given by
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A vase needs to be filled with water. If the vase is a cylinder that is 
tall with a 
radius, how much water is needed to fill the vase?
A vase needs to be filled with water. If the vase is a cylinder that is
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Cylinder
Cylinder
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A cylinder has a base diameter of 12 in and is 2 in tall. What is the volume?
A cylinder has a base diameter of 12 in and is 2 in tall. What is the volume?
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The volume of a cylinder is 
The diameter is given, so make sure to divide it in half.
The units are inches cubed in this example
The volume of a cylinder is
The diameter is given, so make sure to divide it in half.
The units are inches cubed in this example
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A circle has a circumference of 
and it is used as the base of a cylinder. The cylinder has a surface area of 
. Find the volume of the cylinder.
A circle has a circumference of
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Using the circumference, we can find the radius of the circle. The equation for the circumference is 
; therefore, the radius is 2.
Now we can find the area of the circle using 
. The area is 
.
Finally, the surface area consists of the area of two circles and the area of the mid-section of the cylinder: 
, where 
is the height of the cylinder.
Thus, 
and the volume of the cylinder is 
.
Using the circumference, we can find the radius of the circle. The equation for the circumference is
Now we can find the area of the circle using
Finally, the surface area consists of the area of two circles and the area of the mid-section of the cylinder:
Thus,
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What is the volume of a cylinder with a radius of 4 and a height of 5?
What is the volume of a cylinder with a radius of 4 and a height of 5?
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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?
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?
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The volume is the area of the base times the height. The area of the base is 
, and the radius here is 3. 
The volume is the area of the base times the height. The area of the base is
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