Cylinder

To understand a justification for the formula of a cylinder, we must first understand something called Cavalieri's Principle.

  • If two 3D shapes have cross sections of equal area at all heights,

  • and have the same height,

  • then their volumes are equal

What is a "cross section?" Think of it like a sideways slice.

The best picture to imagine is two stacks of coins.

In this picture, the top of each coin is a cross section. So, our cross sections are all circles, since the top of a coin is a circle.

Let's look at the first part of Cavalieri's Principle. Do these figures have equal cross sections at all heights? Yes, because no matter where I look at a cross section, its going to be the area of the circle of a coin, and all the coins are the same.

Let's look at the second part of Cavalieri's Principle. Do these figures have the same height? Yes, both stacks are 10 coins tall.

Then Cavalieri's Principle states that the volumes of both figures are equal. This makes sense, because of course 10 coins would take up the same amount of space no matter how you arrange them.

We can use this idea to prove the formula for the volume of a cylinder. Let's compare a cylinder to a cube.

Here, both shapes have equal heights. Also, the cross sections (the squares and the circles) have equal areas. Then, by Cavalieri's Principle, both of these shapes have equal volumes.

Volume = Base x Height

For the cube, the base is the area of a square, so Base = Length x Width. Substituting into the volume equation, we get

Vcube = Length x Width x Height

For the cylinder, the base is the area of a circle, so Base = π x Radius2. Substituting into the volume equation, we get

Vcylinder = π x Radius2 x Height

Vcylinder = πr2h

Does it matter if the cylinder is slanted? No, because of Cavaliari's Principle

These cylinders both have equal volume

Vcylinder = πr2h