Suppose that you put two objects side by side and sliced them both by a horizontal plane, creating a cross-section of each object.[br][br]If the cross-sections have the same area [i]at every height[/i], then Cavalieri's Principle says that the two objects have the same volume.[br][br]This applet allows you to discover the volume formula for the sphere, using Cavalieri's Principle. [b]Please prepare yourself[/b] by doing these two things:[br][br]1) Know the volume formula for a cone (which is motivated by [url=https://www.geogebra.org/m/bJTpvBrA]this dissection of a cube into pyramids[/url]).[br][br]2) Looking at the "three" objects below, think of the bronze cylinder as one object, and the red cone and purple sphere, [i]combined[/i], as a second object.[br][br]Once you're ready, play and observe closely.[br]

I hope you've made several observations of your own. Did you show the details? Did you twirl a radius or two?[br][br]Here are a few questions that I think you should consider in order to make the most of this applet.[br][br]1. You may have seen the equation (h[sub]s[/sub])[sup]2[/sup]+(r[sub]s[/sub])[sup]2[/sup]=1, which looks kind of like the equation for a circle--but in this context, it actually describes the sphere. How/why?[br][br]2. Can you explain each step of the algebra that appears in the 2-D view when the details are shown?[br][br]3. How are the cross-sectional areas of the cylinder, cone, and sphere related?[br][br]4. What does Cavalieri's Principle tell us about the volumes of the cylinder, cone, and sphere?[br][br]5. Given that the cylinder has radius 1, how tall is it? What is its volume?[br][br]6. What is the volume of the cone?[br][br]7. What is the volume of the sphere?[br][br]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 [url=http://www.cde.ca.gov/be/st/ss/documents/ccssmathstandardaug2013.pdf]California CCSS G-GMD.5[/url].)[br][br]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.[br]