Math Analysis — Example of Analysis (Finding πr²)

Slicing the circle into rings, unrolling into strips — and area πR² via a triangle

How do we find the area of a circle analytically?

Slicing into Rectangles

Idea: slice the circle into thin concentric rings and «unroll» each ring into a rectangle.

A ring of radius r and width Δr has circumference 2πr. Unrolled, it becomes a rectangle:

  • length: 2πr
  • width: Δr
  • area: 2πr · Δr

The total area is the sum of all such rings, from r = 0 to r = R. The smaller Δr, the better the approximation.

Rr=0Rings r = 0 … R

If we arrange these cut strips (each of length 2πr) on a graph from left to right — from r = 0 to r = R — we get a triangle. Base R, height 2πR: the area of the triangle is ½ · R · 2πR = πR².

r2πrR2πR½·R·2πR = πR²

Result: area of the circle S = πR². The same approach — slicing a shape into «small pieces», summing their areas, and taking the limit — underlies integral calculus and, as we will see, leads to derivatives.