M5M1 - Developing Area Formula (8)

M5M1. Students will extend their understanding of area of geometric plane figures.
g. Derive the formula for the area of a circle.

Deriving the area formula for circles is a bit more complex than deriving the formulas for polygons. The fundamental idea, "When you are given an unfamiliar shape, you may be able to calculate its area if you can make a familiar shape (or a collection of familiar shapes)," is still applicable. However, we can't really make a polygons out of a circle. So, the process typically requires an additional step.

However, before we get to the derivation of the area formula, let's think about another idea first. Here is a little puzzle for you. I'm sure many of you have seen racquetball balls sold in a canister like this one:

So, which do you think is loner, the height of the canister or the distance around the canister, or are they about the same? I encourage you to actually find a canister and test it yourself.

Sometimes, they come in a package of 3 balls in a canister like this one:

Again, which is longer, the height or the distance around it, or are they about the same?

Tennis balls usually sold in canisters, sometimes with 3 balls and others may have 4.

Again, which do you think is longer, the height or the distance around it, or are they about the same?

I have once seen a set of 3 softballs being sold in a canister like these, too. Again, which do you think is longer, the height or the distance around it, or are they about the same?

It turns out that whenever there are 3 balls in a canister, the height and the distance around the canister are about the same, no matter what kind of balls they are - the distance around is actually a bit longer but they are pretty close. So, what is happening? A simplified picture of the question will be like this:

In other words, the question was asking you to compare the length around a circle (we call it circumference) and the sum of three diameters. What we observed in the canisters of balls is that the circumference of a circle is about three times of its diameter no matter how big a circle is. In other words, the ratio of the circumference to the diameter is constant, and the value of this ratio is a little more than 3. This ratio is called the ratio of circumference, and we often use the Greek letter, π (pi). This number is an example of irrational number, that is, a number that cannot be expressed exactly as a fraction - you may have seen people use 22/7 as an approximation, but it is not exactly equal to pi. Another, and perhaps more common, approximation of the ratio of circumference is 3.14, and this is mentioned in M5M1h.

What we just observed can be summarized in the following equations:

Because the diameter is twice the radius, we can also express the relationship between the circumference and the radius this way:

It turns out the circumference plays an important role when we are trying to calculate the area of a circle. I will discuss the area of the circle in the next post.