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Surface Area and Volume

Surface Area and Volume

At a Glance - 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, πr2.

V = Bh = πr2h

Sample Problem

What is the volume of this cylinder?

V = πr2h
V = π(6 in)2(18 in)
V = 648π
V ≈ 2035.8 in2

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.

See? If both stacks have the same number of quarters, they've both got the same volume. Even if one of them is all wonky-looking.

So what did Signior Cavalieri's story teach us?

  1. It doesn't matter what you look like on the outside. (Unless you're a Hollywood actor.)
  2. 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?

Thanks to Cavalieri, we know the volume of this bad boy is just the area of the base times the height, no matter how oblique it looks.

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.

a2 + b2 = c2
(9 m)2 + b2 = (15 m)2
b = 12 m

With the height, we can calculate the volume.

V = πr2h
V = π(4 m)2(12 m)
V = 192π
V ≈ 603.2 m3

Cavalieri would be proud.

Example 1

What is the volume of this cylinder if its circumference is 125.7 feet?

Example 2

A cube has a cylindrical hole with a diameter of 5 centimeters cut into it all the way through. What is the volume of the shape?

Exercise 1

What's the volume of a cylinder with a base diameter of 10 mm and a height of 3.5 mm?

Exercise 2

If the total volume of this figure is 788,500 cubic feet, then how tall is the cylindrical tower?

Exercise 3

You built a Pacman-shaped piggybank out of a 1 ft × 1 ft block of wood in your woodshop class. In cubic feet, how much room does your Pacman piggybank have for coins?

Exercise 4

You have 700 jellybeans. (Yes, you counted them all because it was more fun than doing your geometry homework.) You're given a cylindrical cup with a radius of 3 inches and a height of 10 inches. If each jellybean is about 0.5 cubic inches in volume, will you be able to fit all the jellybeans into the cup? If not, how many will be left over?

Exercise 5

This rectangular prism has an oblique cylinder cut out of it. Find the volume of the solid if the diameter of the cylinder is 20 m, the base of the prism is a square with side length 40 m, and the length of the prism is 115 m.

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