I'm having trouble with finding the surface area of a sphere, without using any calculus.
What I thought, was that the surface area of a sphere is fundamentally an infinite number of rings, decreasing in size as you go up or down the actual circumference, stacked up on top each other. By adding the infinite circumferences of the rings up, you should be able to obtain the actual value of the surface area, as by the formula, $4\pi r^2$. The more circumferences you add up in your sum, the closer your value will be to $4\pi r^2$:
If you start with a diameter of 10, and you add up the circumferences of rings with diameters of [1, 2, 3, 4, 5, 6, 7, 8, and 9] x 2 (for the top half of the sphere) + 10 (the actual circumference) you get 100π as the result.
But, if you use the formula for the surface area of a sphere, $4\pi r^2$, you also get 100π as a result!
Although I'm taking the sum of only 19 circumferences of the circle, I am still getting identical values for both ways! Why is this? Shouldn't the sum of the method involving the circumferences of the rings be less than the formula value, and get more and more accurate as i increase the number of rings in my sum?
This doesn't only work for a sphere with a diameter of 10, but for any other sphere as well!