How to Find the Height of a Cylinder

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  • 0:04 Steps to Solve
  • 2:07 Applications
  • 4:19 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

In this lesson, we'll see how to find the height of a cylinder by manipulating the formula for the volume of a cylinder. We'll also look at a practical application of this formula.

Steps to Solve

Before we can learn how to find the height of a cylinder, we need to quickly review some vocabulary. A cylinder is a three-dimensional object with two equal-sized circular ends. These equal-sized circular ends are called bases. The height of a cylinder is the distance between the two circular bases.


heightcyl1


One more vocabulary word, and that is the volume of a cylinder. The volume is the amount of space the cylinder takes up, and we have a nice formula for finding the volume of a cylinder. If V is the volume of a cylinder, h is the height, and r is the radius of the circular bases, then we have the following:

V = πr2h

Or volume is equal to πr2 * height.

At this point, you may be wondering why we are talking about the formula for finding volume of a cylinder when we are interested in finding the height of a cylinder, and that's a fair question! Let's answer it!

The reason we are looking at the volume formula is because we can manipulate this formula to derive a formula for the height of a cylinder.

We begin with:

V = πr2h

If we divide both sides by πr2, we have:

V ÷ (πr2) = h

If we interchange the sides of this equation, we are left with:

h = V ÷ (πr2)

Ta-da! We now have a formula we can use to find the height of a cylinder. That is, if we know the volume, V, of a cylinder and the radius, r, of the circular base of a cylinder, we can find the height, h. We have derived the formula:

h = V ÷ (πr2)

Where V is the volume of the cylinder and r is the radius of the circular bases of the cylinder; so the height of a cylinder is equal to the volume of the cylinder divided by πr2.

heightcyl2

Applications

Okay, time to put this newly-learned knowledge to use. Suppose you have recently inherited a farm that has a huge old silo on it. You decide to fix up the silo so that you can put it back into use. Part of this fixer-upper job involves attaching a ladder to the side of the silo that goes all the way up to the top of the silo.


How Long Should the Ladder Be?
heightcyl3


As you can see, to determine how long the ladder should be, we need to know the height of the silo. You probably also noticed that the silo is in the shape of a cylinder, which is perfect, because we know how to find the height of a cylinder.

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