The Area of a Circle

Triss uses Scott Foresman's Exploring Mathematics currently, and is working like fury to get through the 6th grade book before September so she can begin the University of Chicago School Mathematics Project's Transition Math text (also put out by Scott Foresman).

Today the lesson was an introduction to finding the area of a circle, and I really appreciate the way the authors laid out the explanation of the formula. I do not remember ever coming in contact with this kind of explanation or reasoning for formulas in my own math books when I was in junior high and high school. (I am of the private opinion that I might have been a math person had I had the right math teachers in school-- there is so much about math that pleases me when I finally get concepts. I coulda been a contendah! But I digress.)

I want to share their explanation:

First off, Ivey had to make a circle with a radius of four inches out of construction paper. Then she drew eight diameters through it at regular intervals. After that she cut it into "pie slices," for lack of a better description, and ended up with sixteen slices of circle. Then she was instructed to lay the slices out in a row, alternating top to bottom, so that they made a rough parallelogram.

Now here is the cool part.

They had her measure the base of the "parallelogram" and compare it to the circumference of the circle (at this point she wanted my help, and we got roughly half the circumference). Then they had us measure the height of the "parallelogram" and compare it to the radius of the circle. They were equal. Then they led us through the reasoning process that begins with the formula for the area of parallelograms, and ends with the formula for a circle. I really got excited at this point because I had never realized those two formulas connected. Here is what it looked like (I'm writing it in words below each equation to help non-math folks-- me-- understand just how cool this is):

Area = base x height
(The area is equal to the base plus times the height)

Area = (1/2 x C) x r
(Since the base and height of a circle is very difficult to measure, we have to take the circle apart and make it into something easier, ie., a parallelogram. And look! the base of the resulting parallelogram is equal to half the circumference, while the height is equal to the radius. This means that within the equation for area, we can substitute half the circumference for the base and the radius for the height. Then we won't half to continue cutting circles into pie slices and fitting them into paralellograms each time.)

A = 1/2 x (2 x pi x r) x r
(But that is a cumbersome equation because we still have to figure out the Circumference before figuring out the area. So let's take the equation for circumference, which is 2 x pi x r, combine it with the rest of the equation, and simplify it.)

A = pi x r(squared)
(Since 1/2 x 2 equals 1, and 1 times anything is equal to the anything, we can cancel out the 1/2 and the 2. Then we have pi x r x r. r x r is the same as saying "radius squared", so let's just say that. Voila!)

Pi times radius squared is the formula for circles I know from school, but I never learned how "they" decided it was the magic formula that would always result in the area of a circle. To have a math book appreciate the innate intelligence of regular students enough to explain where a formula came from is inspiring. It helps a person begin to realize that perhaps math is accessible after all, though it is also challenging.

(And yes, I know I am a geek. But I like it.)