High School: Geometry
Geometric Measurement and Dimension HSG-GMD.A.2
2. Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.
From ecosystems to the inner lives of cells, biology has instilled fear in middle and high school students all across the world. No activity caused more dread, however, than the inevitable biology dissection, which can send animal rights activists into a fury and queasy students into unconsciousness.
Fortunately, dissections in math are far less controversial (but only a little less messy).
Students should be familiar with the basic concept of Cavalieri's principle, especially when applied to oblique solids. What they might not know is that we can use Cavalieri's principle to find the volume of a sphere. We'll start with a cylinder with radius r and a height of 2r. Inside, let's put two cones, each with a height and a radius of r.
If we pass a plane through the top cone such that the inner circle has a radius of b, we can calculate the area of the shaded ring as π(r2 – b2).
Now let's look at our sphere of radius r.
The cross section that is b up from the great circle. We can call the radius of the cross section circle x and make a right triangle with legs b and x and hypotenuse r. We know that x2 + b2 = r2 (thanks, Pythagoras), or x2 = r2 – b2. If we multiply both sides by π, we should get the cross section's area, A = πx2 = π(r2 – b2).
Look familiar? This is the same as the area of our ring from above. And since we would get the same thing for any b, then according to Cavalieri's principle, the volume of our sphere is equal to the volume of the solid between the cones and our cylinder which means V = 2πr3 – ⅔πr3 = 4⁄3πr3.
Students can follow a similar process for other regular solid figures. For example, can they apply Cavalieri's principle to find the formula for the volume of a regular tetrahedron?
Working through some of these derivations should help students better understand the volume formulas. These volume formulas didn't come out of nowhere, and knowing how to derive them will not only give them a more solid (pun intended) understanding of volume, but it will also make memorizing these formulas unnecessary. If they're ever stuck, they'll have the tools to derive the formula instead.
Students might complain about how messy these derivations are, but it's nowhere near as bad as a cutting a frog open. Yuck.