How to find the volume of a cylinder - Intermediate Geometry

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Question

If a cylinder has a radius, $\small r$ , of 2 inches and a height, $\small h$ , of 5 inches, what is the total surface area of the cylinder?

Answer

The total surface area will be equal to the area of the two bases added to the area of the outer surface of the cylinder. If "unwrapped" the area of the outer surface is simply a rectangle with the height of the cylinder and a base equal to the circumference of the cylinder base. We can use these relationships to find a formula for the total area of the cylinder.

$A=2A_{base}+A_{rectangle}$

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

Use the given radius and height to solve for the final area.

$\small 2\pi(2)^{2} + 2\pi (2)(5)$

$\small 8\pi + 20\pi$

$\small 28\pi$

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Question

The volume of a cylinder is $64\pi$ , what is its height?

Answer

Because both the radius and height are unspecified, any real number could be it's height as long as an matching radius is also chosen. There is no restriction on the height or radius being rational.

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Question

What is the volume of a hollow cylinder with an outer diameter of , an inner diameter of and a length of ?

Answer

The general formula for the volume of a hollow cylinder is given by $V=\pi (r_{o}^{2}-r_{i}^{2})l$ where $r_{o}$ is the outer radius, $r_{i}$ is the inner radius, and is the length.

The question gives diameters and we need to convert them to radii by cutting the diameters in half. Remember, . So the equation to solve becomes:

$V=\pi (4^{2}-1^{2})18$ or $V=270\pi; in^{3}$

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Question

Find the volume of the following right cylinder:

Answer

The correct answer is $160\pi \ in^{3}$ .

The formula for volume of a cylinder is

$\pi r^{2} h$

and

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Question

Given a cylinder with radius of 5cm and height of 10cm, what is the volume of the cylinder?

Answer

The volume of a cylinder is given by $\pi r^2\cdot height$

Notice how the formula for the volume is defined as the area of a circle times the lateral height of the cylinder. It is as if we are taking little paper circles and stacking them one-by-one until we fill up the entire container.

Plugging in the numbers we get:

$\pi (5)^2\cdot 10 = 250\pi$

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Question

A right cylinder has a diameter of and a height of . What is the volume of this cylinder?

Answer

The formula to find the volume of a cylinder is: $V = \pi \cdot r^2 \cdot h$ , where is the radius of the cylinder and is the height of the cylinder.

A good point to start in this kind of a formula-based problem is to ask "What information do I have?" and "What information is missing that I need?"

In this case, the problem provides us with the height of the cylinder and its diameter. We have the component of the equation, but we're missing the component. Can we find out ? The answer is yes! Radius is half of diameter. So in this case, because the diameter is , the radius must be .

Now that we have and , we are ready to solve for the volume after substituting in those values.

$V = \pi \cdot r^2 \cdot h$

$V = \pi \cdot (1)^2 \cdot 10$

$V = \pi \cdot \1 \cdot 10$

$V = 10 \pi: in^3$

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Question

You have just bought a farm that includes a storage silo. You want to find the volume for the 20 foot tall silo in the shape of a cylinder. You measure the circumference of the silo, it is listed below.

$Circumference=10\pi\ ft$

Leave answer in terms of $\pi$ .

Answer

To find the volume of a cylinder we need to find the area of the base and then multiply that by the height. The base is in the shape of a circle so the formula is given below:

$V=\pi\ r^{2}\ h$

To find the radius of the base, we use what was found from the measurement for circumference of the base. We set the formula for circumference equal to the measured circumference given in the problem.

$2\pi\ (r)=10\pi$

From this we can solve to find that the radius =5. Which we can plug into the original Volume formula. With the given height of 20 ft.

$V=\pi\ 5^{2}\ (20)$

Simplifying will give us the volume of the cylinder.

$V=500\pi\ ft^{3}$

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Question

A can of tomato soup has a diameter of 3.5 inches and a height of 8 inches. What is the volume of the can of soup?

Answer

To find the volume of a soup can, or a cylinder, we use the formula for volume:

$\text {Base Area x Height = Volume}$

The base of our cylinder is a circle and to find the area of the circle we use:

$\text {Area =} ; \pi\times r^2 ;, \text {here our radius = }\frac{3.5}{2} = 1.75cm$

Now we can use the formula for the area of our circular base:

$\text {Area =} ; \pi\times (1.75)^2 = 3.0625\pi ; cm^2$

All we need to do now is multiply the base by the hieght of our cylinder!

$3.0625 \pi ; cm^2 \times 8cm = 24.5\pi ; cm^3$

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Question

You are painting a water tank black. You are trying to find the surface area of the tank so you measure the height and diameter of the tank. The height is 2.5 meters, and diameter is 4 meters. What is the total surface area of the tank?

Answer

To find the surface area of a cylinder we need to find the area of the circular top and bottom and the area of the rectangular (but rounded) side, then add them together. Lets start by finding the area of the circular ends to the cylinder, remember that we measured the diameter (d) and we need the radius (r) to find the area of a circle, the radius is half of the diameter:

$A = r^2\pi = 2^2 \pi = 4\pi ; m^2$ (area of one circle)

Don't forget that we have two circles on the cylinder that have this area:

$4\pi ; m^2 + 4\pi ; m^2 = 8\pi ; m^2$ (area for both circles)

Now we just need to find the area of rectangular side of the cylinder. To do this we will take the hieght of the cylinder, 2.5 m, and multiply by the circumference (C) of the circle (the width of the rectangular side of the cylinder).

$C = 2\pi r ; \text{or} ; C= d\pi = 4\pi ; cm$

So now that we know the circumference we can multiply by the cylinder height to find the area of the rectangular side:

$4\pi ; cm \times 2.5 ; cm = 10\pi ; cm^2$ (area for rectangular side)

Now just add to find the total surface area:

$8\pi ; cm^2 + 10\pi ; cm^2 = 18\pi ; cm^2$

and that is our total surface area for the cylinder!

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Question

Find the volume of a cylinder that has a radius of and a height of .

Answer

Recall how to find the volume of a cylinder:

$\text{Volume of Cylinder}=\text{Area of base}\times\text{height}$

Since the base of a cylinder is a circle, we can write the following equation:

$\text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$

Substitute in the given values to find the volume.

$\text{Volume of Cylinder}=\pi\times 2^2\times 4$

Solve.

$\text{Volume of Cylinder}=16\pi$

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Question

Find the volume of a cylinder that has a radius of and a height of .

Answer

Recall how to find the volume of a cylinder:

$\text{Volume of Cylinder}=\text{Area of base}\times\text{height}$

Since the base of a cylinder is a circle, we can write the following equation:

$\text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$

Substitute in the given values to find the volume.

$\text{Volume of Cylinder}=\pi\times 3^2\times 3$

Solve.

$\text{Volume of Cylinder}=27\pi$

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Question

Find the volume of a cylinder that has a radius of and a height of .

Answer

Recall how to find the volume of a cylinder:

$\text{Volume of Cylinder}=\text{Area of base}\times\text{height}$

Since the base of a cylinder is a circle, we can write the following equation:

$\text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$

Substitute in the given values to find the volume.

$\text{Volume of Cylinder}=\pi\times 4^2\times 2$

Solve.

$\text{Volume of Cylinder}=32\pi$

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Question

Find the volume of a cylinder that has a radius of and a height of .

Answer

Recall how to find the volume of a cylinder:

$\text{Volume of Cylinder}=\text{Area of base}\times\text{height}$

Since the base of a cylinder is a circle, we can write the following equation:

$\text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$

Substitute in the given values to find the volume.

$\text{Volume of Cylinder}=\pi\times 5^2\times 4$

Solve.

$\text{Volume of Cylinder}=100\pi$

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Question

Find the volume of a cylinder that has a radius of $\frac{1}{2}$ and a height of .

Answer

Recall how to find the volume of a cylinder:

$\text{Volume of Cylinder}=\text{Area of base}\times\text{height}$

Since the base of a cylinder is a circle, we can write the following equation:

$\text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$

Substitute in the given values to find the volume.

$\text{Volume of Cylinder}=\pi\times\left( \frac{1}{2}\right)^2\times 8$

Solve.

$\text{Volume of Cylinder}=2\pi$

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Question

Find the volume of a cylinder that has a radius of and a height of .

Answer

Recall how to find the volume of a cylinder:

$\text{Volume of Cylinder}=\text{Area of base}\times\text{height}$

Since the base of a cylinder is a circle, we can write the following equation:

$\text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$

Substitute in the given values to find the volume.

$\text{Volume of Cylinder}=\pi\times 6^2\times 2$

Solve.

$\text{Volume of Cylinder}=72\pi$

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Question

Find the volume of a cylinder that has a radius of and a height of .

Answer

Recall how to find the volume of a cylinder:

$\text{Volume of Cylinder}=\text{Area of base}\times\text{height}$

Since the base of a cylinder is a circle, we can write the following equation:

$\text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$

Substitute in the given values to find the volume.

$\text{Volume of Cylinder}=\pi\times 15^2\times 9$

Solve.

$\text{Volume of Cylinder}=2025\pi$

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Question

Find the volume of a cylinder that has a radius of $\frac{1}{4}$ and a height of .

Answer

Recall how to find the volume of a cylinder:

$\text{Volume of Cylinder}=\text{Area of base}\times\text{height}$

Since the base of a cylinder is a circle, we can write the following equation:

$\text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$

Substitute in the given values to find the volume.

$\text{Volume of Cylinder}=\pi\times \left(\frac{1}{4}\right)^2\times 48$

Solve.

$\text{Volume of Cylinder}=3\pi$

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Question

Find the volume of a cylinder that has a radius of and a height of .

Answer

Recall how to find the volume of a cylinder:

$\text{Volume of Cylinder}=\text{Area of base}\times\text{height}$

Since the base of a cylinder is a circle, we can write the following equation:

$\text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$

Substitute in the given values to find the volume.

$\text{Volume of Cylinder}=\pi\times 13^2\times 144$

Solve.

$\text{Volume of Cylinder}=2028\pi$

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Question

Find the volume of a cylinder that has a radius of and a height of $\frac{1}{8}$ .

Answer

Recall how to find the volume of a cylinder:

$\text{Volume of Cylinder}=\text{Area of base}\times\text{height}$

Since the base of a cylinder is a circle, we can write the following equation:

$\text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$

Substitute in the given values to find the volume.

$\text{Volume of Cylinder}=\pi\times 2^2\times \frac{1}{8}$

Solve.

$\text{Volume of Cylinder}=\frac{1}{2}\pi$

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Question

Find the volume of a cylinder that has a radius of and a height of $\frac{1}{4}$ .

Answer

Recall how to find the volume of a cylinder:

$\text{Volume of Cylinder}=\text{Area of base}\times\text{height}$

Since the base of a cylinder is a circle, we can write the following equation:

$\text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$

Substitute in the given values to find the volume.

$\text{Volume of Cylinder}=\pi\times 18^2\times \frac{1}{4}$

Solve.

$\text{Volume of Cylinder}=81\pi$

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