Have you ever wondered where the formula \(A = \pi r ^ 2\) comes from?
The explanation is quite simple, and is based somewhat on Calculus concepts like limits.
Take a circle, like the one above. The radius is \(r\) and circumference \(2\pi r \). How could one figure out the total space inside it easily, without trying to fit tons of \(n\)-gons in it like Archimedes? We can't use squares or polygons because they are not totally accurate. They leave out space on the edge. But who said we have to work with shapes? Let's work with lines; more specifically, curves, like the one we are given.
What if you divided the circle up into rings, infinite amounts of them, so that they covered the entire circle? Starting with the circumference as the first ring and working inwards, the rings get constantly and gradually shorter. Thinking of each ring as the circumference of a smaller and smaller circle, the radius is getting smaller proportionally to the circumference. Eventually, we get to a point in the middle, a ring with 0 length. Now that we have this infinite amount of lines, you may ask, what can we do with these? How can we turn 1-dimensional lengths into a 2-dimensional area?
The answer in part 2.