Volume of a Sphere via Cavalieri's Principle
- Brad Ballinger
Cavalieri's Principle: Background information
Suppose that you put two objects side by side and sliced them both by a horizontal plane, creating a cross-section of each object. If the cross-sections have the same area at every height, then Cavalieri's Principle says that the two objects have the same volume. This applet allows you to discover the volume formula for the sphere, using Cavalieri's Principle. Please prepare yourself by doing these two things: 1) Know the volume formula for a cone (which is motivated by this dissection of a cube into pyramids). 2) Looking at the "three" objects below, think of the bronze cylinder as one object, and the red cone and purple sphere, combined, as a second object. Once you're ready, play and observe closely.
I hope you've made several observations of your own. Did you show the details? Did you twirl a radius or two? Here are a few questions that I think you should consider in order to make the most of this applet. 1. You may have seen the equation (hs)2+(rs)2=1, which looks kind of like the equation for a circle--but in this context, it actually describes the sphere. How/why? 2. Can you explain each step of the algebra that appears in the 2-D view when the details are shown? 3. How are the cross-sectional areas of the cylinder, cone, and sphere related? 4. What does Cavalieri's Principle tell us about the volumes of the cylinder, cone, and sphere? 5. Given that the cylinder has radius 1, how tall is it? What is its volume? 6. What is the volume of the cone? 7. What is the volume of the sphere? 8. In your mind or on paper--not in the applet--scale everything up by the factor r. In other words: make the sphere have radius r instead of 1; make the cylinder large enough to contain the sphere but no larger; create the double cone to have the same two bases that the cylinder has. Now what are the three volumes? (You can either redo everything from scratch, or one-step it with California CCSS G-GMD.5.) If everything has gone according to plan, you now have not only the formula for the volume of a sphere, but also a way of thinking about it that will help you (1) explain why that formula is correct, and (2) reconstruct that formula if you forget it.