The Area of a Circle

Farmer Johnson is bummed out. Crop circles are destroying his farm, and the bandits are making off with his corn! He's not saying it was aliens, but probably it was aliens. How can Farmer Johnson recover from this devastating loss? It's Insurance Agent Frank to the rescue! Frank explains that Farmer Johnson needs to calculate the area of the farm that was destroyed, and his insurance company will compensate him for his loss! To do that, Farmer Johnson will need to find the area of circles. Unfortunately, Johnson doesn't know the formula, but he does have a delicious looking corn pie! No, not to eat. Maybe looking at the pie, which is also a circle, can help him find the formula he needs. Farmer Johnson does know the formula for the area of a rectangle, so he thinks about how that could help him. Let's see what happens when we circumscribe a square around the circle. Since a square is a special kind of rectangle, we know how to calculate the area of the square. But this approximation is quite a ways off from the area of the circle. And we don't have a way to calculate the area between the circle and square, so it looks like we're stuck. Let's try something else. What about putting unit squares INSIDE the circle? If we knew how many unit squares fit inside, then we could add up their areas to get the total area of the circle. Counting squares works in the interior, but at the edges, many are only partially filled. This could only gives us a rough approximation, which isn't good enough for Farmer Johnson. Farmer Johnson is a bit frustrated and decides to have some pie. He slices it into 12 equal pieces, cutting through the center. This length here is the radius, the distance from the center of the circle to the edge of the circle. The circumference is the length around the entire circle, and this section here is half of that, so its length can be written as 'C' over 2. Hold on a second! Johnson's got an idea, and it's so crazy it just might work! These slices can be fit together with half of this piece moved to the other side, and it looks similar to a rectangle. If we can find the exact or approximate area of this rectangle, we will know more about the area of the circle. To find the area of this rectangle, what 2 measurements do we need to know? Its length and its width. The width of the rectangle is approximately equal to half of the circumference of the original circle. The circumference is given by the formula 2 pi 'r', so the width is half of that: pi times 'r'. What about the length of the rectangle? The length of the rectangle is approximately equal to the length of each slice, which is 'r'. Do you remember the formula for the area of a rectangle? It's area equals length times width. Substituting 'r' and pi 'r' into the formula gives us the area of the rectangle equal to approximately pi 'r' squared. If we divided the circle into more, smaller pieces, then the area of the circle would get closer and closer to being the area of the rectangle. As the number of pieces reaches infinity, it would be exactly equal to the rectangle. That means the area of a circle equals pi times the radius squared! Farmer Johnson is ready to use his formula for the area of a circle. One of the crop circles has a radius of 10 feet. Plugging this into our formula tells us that the area is 'pi' times ten squared. That's 100 pi feet squared. Since pi is about 3.14, the area is about three hundred fourteen square feet. Johnson calculates the areas of the bigger crop circles as well, and sends the results to Agent Frank. While he waits for a reply, let's review how we found the area of a circle. We divided the circle into equal pieces, and then rearranged them into a shape that resembled a rectangle. By finding the area of this rectangle approximation, we discovered the area of a circle. The formula is area equals pi times the radius squared. If we are given the radius, we can simply plug it into the formula. If we are given the diameter or circumference instead, we first have to find the radius and then plug it into the formula. Agent Frank gives Farmer Johnson his check, saving the day once again. Johnson is happy as a pig in mud! But there's one thing still stuck in his craw: Was it actually ALIENS? And what did they want with all that corn anyway?