# Surface Area and Volume

# Volume of Cylinders

The volume formula works not only for prisms, but for cylinders, too. The only difference is that the base is circular instead of triangular or rectangular or whatevular. If that's true (which it totes is), we can substitute the area of the base *B* for the area of a circle, π*r*^{2}.

*V* = *Bh* = π*r*^{2}*h*

### Sample Problem

What is the volume of this cylinder?

*V* = π*r*^{2}*h**V* = π(6 in)^{2}(18 in)*V* = 648π*V* ≈ 2036 in^{2}

Voluminously victorious.

Up until now, we've been dealing with right prisms and right cylinders. What about their oblique buddies? They probably feel a little left out.

Don't worry, obliques. Just because you aren't right doesn't mean you're wrong. We can calculate your volumes just as easily as those goody-goody two-shoes right cylinders, thanks to an Italian mathematician named Signior Cavalieri.

*The Life and Legend of Signior Bonaventura Cavalieri*

Bonaventura (but he goes by Bonnie) Cavalieri was born in Italy over 400 years ago as an oblique cylinder. All throughout school, his right cylinder classmates teased him for being different, so he tried whatever he could to straighten out his obliqueness and be like everyone else. Of course, that didn't work, so Cavalieri devoted his life to proving that he was made of the same stuff as any other right cylinder.

One day while making a sandwich, his knife slipped and he cut himself in half. (Don't worry. He's a cylinder. He just super-glued himself back together.) He realized that it didn't matter how the circles were arranged in a cylinder. All that mattered was that the circles were all the same.

Cavalieri named a principle (not a principal) after himself that said that as long as two cylinders have the same height and cross-sectional area at every level, they had the same volume. He proved it using two stacks of coins (after gluing himself back together, of course). You show 'em, Cavalieri.

So what did Signior Cavalieri's story teach us?

- It doesn't matter what you look like on the outside. (Unless you're a Hollywood actor.)

- If the height and radii for two cylinders are the same, their volumes are the same regardless of their rightness or obliqueness.

As tribute to Cavalieri and the difficult life he faced as an oblique cylinder, we'll calculate the volume of one.

### Sample Problem

What is the volume of this cylinder?

Since we only have the slant height and we want the actual height, it's pretty clear what we need to do. Especially with that right angle staring us in the face.

*a*^{2} + *b*^{2} = *c*^{2}

(9 m)^{2} + *b*^{2} = (15 m)^{2}*b* = 12 m

With the height, we can calculate the volume.

*V* = π*r*^{2}*h**V* = π(4 m)^{2}(12 m)*V* = 192π*V* ≈ 603 m^{3}

Cavalieri would be proud.