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The area of a circle
You know that the circumference of a circle is . You need to find the area. What should you multiply the circumference by?
This is a pancake, how big is it? What is it's area? Calculating the area of a circle is a problem that puzzled all early human civilizations. One of the great mathematicians, Archimedes of Syracuse, came up with a way to do it. If you divide a circle into smaller and smaller pieces, you can eventually make a rectangle out of it, and the area of a rectangle is easy to calculate.
Do this, start by dividing the circle into quarters, and put the four sectors of the circle like this, like a rectangle. Well, it doesn't look quite like a rectangle. Divide again so you get eight sectors, and again to get 16 sectors. If you move half a piece like this, it starts to look like a rectangle. If you continue to divide it, the pancake will eventually turn to mush.
But you can see the pattern. Every time you divide it, the shape looks more and more like a rectangle. The wavy base of a rectangle gets straighter and straighter as you make smaller sectors. An infinitely divided pancake is a perfect rectangle. Can you see what happens with the pancake's circumference when you divide? It straightens out and half of it ends up at the top of the rectangle and the other half at the bottom.
The rectangle's base is therefore half the circle's circumference, and that's good to know when calculating the area. But the height of the rectangle, where does that come from? Look at the cuts you made in the pancake. They are the circle's radius, and those cuts gets straighter and straighter until they are perpendicular to the base. This rectangle's height is therefore the circle's radius. So now we have both the base and the height, and the rectangle's area is the base times the height, B x H.
The base of this rectangle is half the circle's circumference, O / 2. The height of the rectangle is the circle's radius, R. When Archemedes came this far, the next step was easy, because he had already managed to calculate the value of pi to two decimal places. Pi is the ratio of the circle's circumference to its diameter. Keep up now.
The circumference is the diameter times pi. Replace O with D x pi, and the diameter is twice the radius. Replace D with 2 x R. Now we can simplify. 2 / 2 is 1, so this goes away.
And R x R is R squared. We don't need to write the multiplication sign. A circle's area is therefore pi x R squared. Did you get that? Take another look if you are lost.
Pay attention to how the circle's radius becomes the height, and how half of the circumference becomes the base. Now we can test the formula. Here is a circle with the diameter of 28 meters - a swimming pool perhaps? How do we calculate the pool's area? We use the formula, the area of a circle equals pi times the radius squared. Pi is approximately 3,14, and the radius is half the diameter - 28 / 2.
Now we just use a calculator. Feel free to pause the video. Here, you have to know the order of operations. First the parentheses, then the exponents, and last, the multiplication. The area is 615,44 square meters.
Since we measured the diameter in meters, we get the answer in square meters. You can also calculate the area with more decimals in pi; you'll notice that the answer will be a little larger. You only need to know one thing by heart to be able to do this: The area of a circle equals pi times the radius squared. If this video was hard to understand, watch the videos about the circumference of a circle, the area of a rectangle, and irrational numbers. All of these things are related.