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:# Comments

The interesting thing about the first two proofs is that between them they interchange the radius r and the circle constant τ in the bounds versus the integrand. The interesting thing about the last demonstration is that it matches the phrasing of Archimedes' original, pre-algebraic conclusion: the area of a circle is equal to that of a triangle with height equal to the radius, and base equal to the circumference (proven with more formal geometric methods).Of course, if you replace τ in the foregoing with 2π, then the multiply-and-divide by 2's cancel out, and you get the more familiar expression πr². But I actually like seeing the factor of r²/2 in the version with τ, as both foreshadowing and reminder that ∫ r dr = r²/2, and also that it's half of a certain rectangle (as I might paraphrase Archimedes). In addition, Michael Hartl makes the point that this matches a bunch of similar basic formulas in physics (link).

Thanks to MathCaptain.com and Wikipedia for the images above.

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