# Intermediate Geometry : How to find the volume of a cylinder

## Example Questions

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### Example Question #1 : How To Find The Volume Of A Cylinder

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

Explanation:

The volume of a cylinder is given by

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:

### Example Question #1 : How To Find The Volume Of A Cylinder

The volume of a cylinder is , what is its height?

Any positive rational number

Any positive real number

Any positive real number

Explanation:

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.

### Example Question #1 : How To Find The Volume Of A Cylinder

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

Explanation:

The general formula for the volume of a hollow cylinder is given by where is the outer radius, 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:

or

### Example Question #1 : How To Find The Volume Of A Cylinder

Find the volume of the following right cylinder:

.

.

Explanation:

The formula for volume of a cylinder is

and

### Example Question #1 : How To Find The Surface Area Of A Cylinder

If a cylinder has a radius, , of 2 inches and a height, , of 5 inches, what is the total surface area of the cylinder?

Explanation:

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.

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

### Example Question #1 : How To Find The Volume Of A Cylinder

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

Explanation:

The formula to find the volume of a cylinder is: , 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.

### Example Question #1 : How To Find The Volume Of A Cylinder

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.

Explanation:

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:

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.

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.

Simplifying will give us the volume of the cylinder.

### Example Question #1 : How To Find The Volume Of A Cylinder

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?

Explanation:

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:

(area of one circle)

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

(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).

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

(area for rectangular side)

Now just add to find the total surface area:

and that is our total surface area for the cylinder!

### Example Question #1 : How To Find The Volume Of A Cylinder

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?

Explanation:

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

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

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

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

### Example Question #28 : Cylinders

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

Explanation:

Recall how to find the volume of a cylinder:

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

Substitute in the given values to find the volume.

Solve.

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