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Where does the formula for the area of a circle come from?

Updated: Jan 1, 2021

Where does the formula for the area of a circle come from? - By Felicia

You probably will/did memorize the formula for the area of a circle in middle school: πr2. But how does it work and where did it come from? This article will explain to you how this formula came to be.

First of all, how does it work? Let’s use an example from Khan Academy. A candy machine creates small chocolate wafers in the shape of circular discs. The diameter of each wafer is 16 mm. That’s all the information you have. Now, let’s get solving. Our formula is πr2. In other words: pi x radius x radius. Since we are given the diameter, 16 mm, all we have to do is divide it by 2, which makes the radius 8 mm. Let’s simplify the equation: pi times 8 squared, or pi times 64. So, the answer is 64π. But sometimes, we might want a more satisfying answer. In that case, pi can be rounded to 3.14, so it’s 3.14 x 64. 3.14 x 64 equals 200.96, and that’s the answer.

Second of all, where did it come from? From my research, it was Archimedes who discovered this formula. Archimedes' essay on the “Measurement of a circle” is often referred to but little read. What most people are interested in is the prescription for approximating pi by considering polygons with a large number of sides. Archimedes' essay comes in two parts. The first is a statement and proof of a relationship between the area of a circle and its circumference. This part is the one we are interested in. The opening statement of this part is this: “The area of any circle is equal to a right-angled triangle in which one of the sides about the right angle is equal to the radius, and the other to the circumference of the circle.” Wait, whaaaaat? Let me explain.

First, let’s say that we have a circle A. I already divided it into 8 pink triangles like so. Then, I placed the pink triangles flat, so that their points are facing up. Now, each triangle has a name: A1, A2, A3, and so on. Adding up all eight triangles gives us the whole circle A. Now, I know this gives us an octagon, but we have to start with 8 pieces. Calculating the area of eight triangles is not the best option. What we can do is calculate the area of a rectangle. Doubling the first eight triangles, shown in yellow here, turns the figure into a rectangle. Since the first 8 triangles or the whole circle was doubled, we must cut the rectangle in half to find the area of one circle. Finally, we have to clean it up. The last image shows it clearly: one pink triangle where one edge is equal to the radius and another edge is equal to the circumference. Ta-da! But wait. This doesn’t really work! The eight triangles make an octagon, not a circle. That’s where we bring in infinity. Infinite triangles will create the perfect pink triangle that is shown below.

We then have to actually find the area of the triangle, which equals the area of the circle. The formula for the area of a triangle is base x height divided by 2. So, radius x circumference divided by 2, right? Yes, but that’s not πr2. Well, actually, it is. See, let’s go back to the rectangle. The formula for area of a rectangle is base x height. Base is circumference, height is radius. The formula for the circumference is π x diameter, or 2πr. 2πr x radius equals 2πr2. 2πr2 is the area of the rectangle. Now, time to cut the rectangle in half! Cutting the rectangle in half is also known as dividing by 2. 2πr2 divided by 2 = πr2, and that’s the answer.

P. S. If this explanation was not enough/too confusing for you, don’t worry. Sooner or later, I will post an interactive demonstration proof for πr2 using cardboard, paint, and glue. See you next time!

P. P. S. πr2 means "pi x radius squared".

P. P. P. S. If you're wondering what the second part of Archimedes' essay on the “Measurement of a circle” was, it was about pi and how it came to be. If you want to learn more about this, I will soon post another interactive demonstration proof for π! Click here to watch it.

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