Common Core Standards: Math
- The Standard
- Sample Assignments
- Practice Questions
1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.
What is love? (Baby, don't hurt me.) It's a question that has haunted humankind for centuries. Scientists, artists, and philosophers have all tried to find the answer, so maybe it's time for mathematicians to take a crack at it.
Like geometry, love spans across all dimensions. Like a circle, it continues forever and forever. Like volume, it fills us up to the brim with happiness. So naturally, if your students want to understand love, they have to understand the various dimensions when it comes to circles. (Okay, we admit that last one was a stretch.)
The first dimension of circles includes radius, diameter, and circumference. Students should easily see the circumference of a circle as a special type of perimeter. Students should also understand that the ratio of the circumference to the diameter is always equal to the yummiest number: π.
If they need help visualizing this, they should cut a piece of string at the length of the circumference and compare it with the measurement of the diameter. The ratio of circumference to diameter should equal π (about 3.14). That's where we get the formula C = πd = 2πr.
For the second dimension, order a pizza. Pepperoni, cheese, veggie, whatever you want. The messier, the better, we say. If we cut the pizza up into 8 slices and line them up in a row, one up, one down, we can create something that looks more or less like a lumpy parallelogram. Looks kind of funky, but no less delicious.
If we cut the pizza into infinitely small slices, the "waves" on the top and bottom of these "parallelograms" would essentially disappear. The base of the parallelogram would be half the circumference of the circle (πr) and the height would be r. If we substitute these values into the area of a parallelogram A = bh, we'd end up with the area of a circle: A = πr2.
You thought the authentic Italian pizza would satisfy your appetite, but it's only made you hungrier. How about an all expense paid trip to Italy? (Sign us up.)
To examine volume, we can look at the Leaning Tower of Pisa for some help. If we built the Upright Tower of Pisa (with the exact same height as the original), both towers would have the same volume. That's because all horizontal cross sections of both towers will be identical at any given height.
Students should understand this concept as Cavalieri's principle, and be able to extend it to other solid figures. (No, this concept doesn't only apply to the Tower of Pisa.) It also helps if students know the volume formulas for prisms, cylinders, cones, pyramids, and spheres.
Perhaps love has its roots in Italy. After all, Romeo and Juliet were from Italy, Italian is the most romantic of the romance languages, and we just explained all three dimensions of geometry using pizza and Pisa. And now that we think of it, Italians have explained love in two simple words: that's amore.
A sample video to help students with surface area: