Saturday, March 14, 2015

Pi Proofs for the Area of a Circle

For Half-Tau Day: Approximately π short proofs for the area of a circle, each in terms of τ = 2π (that is, one "turn"),  mostly using calculus.

Shells in Rectangular Coordinates

See the first picture above. Imagine slicing the disk into very thin rings, each with a small width ds, at a radius of s, for a corresponding circumference of τs (by definition of τ). So each ring is close to a rectangle if straightened out, with length τs and a width of ds, that is, close to an area of τs ds. In the limit for all radii 0 to r, this gives:

Sectors in Polar Coordinates

See the second picture above. Imagine slicing the disk into very thin wedges, each with a small radian angle dθ and a corresponding arclength of r dθ (by definition of θ). So each wedge is close to a triangle (half a rectangle) with base r and a height of r dθ, that is, close to an area of r²/2 dθ. In the limit for all angles 0 to τ, this gives:

Unwrapping a Triangle

Sort of taking half of each of the ideas above, we can geometrically "unwrap" the rings in a circle into a right triangle. One radius stays fixed at height r. The outermost rim of the circle becomes the base of the triangle, with width as the circumference τr. So the area of this triangle is: