PROOF 50

Prove.

The area of a circle is πr²

On the Euclidean plane.

What you may use

A1pₙ → 2πr, qₙ → 2πr
circumference 2πr — the common limit of inscribed and circumscribed regular n-gon perimeters, and the definition of π
A2area of a triangle = ½bh
area of a triangle
A3non-overlapping ⟹ areas add
additivity of area
A4X ⊆ Y ⟹ [X] ≤ [Y]
monotonicity of area
A5the disk D has an area K
existence of area
A6in a right triangle a²+b²=c²
Pythagorean theorem (only cited here)
A7tangent ⊥ radius at the point of tangency
Euclid's Proposition III.18
A8the disk is convex
the segment joining two points of the disk lies in the disk
A9the circle lies on one side of a tangent
a tangent meets the circle only at the point of tangency (Euclid's Proposition III.16)
A10isosceles triangle: the perpendicular bisects
the foot of the perpendicular from the apex to the base is the midpoint of the base (SSS congruence)
A11lim(xy) = lim x · lim y
arithmetic of limits — sums, differences, products
A12xₙ≥0, xₙ²→L² ⟹ xₙ→L
continuity of the square root
A13aₙ ≤ K ≤ cₙ, both →L ⟹ K=L
squeeze theorem
A14associativity, commutativity, distributivity, cancellation
arithmetic of the reals
A151/n → 0
Archimedean property
Ea = b ⟹ same operation on both sides
properties of equality

Nothing outside this list may be cited.

Definition D1

On the circle with center O and radius r, take the points dividing the circle into n equal parts (n ≥ 3). Let Iₙ be the regular n-gon with those points as vertices, and Cₙ the regular n-gon formed by the tangents drawn at those points.

Proof.

Related labs

Perimeter & Area lab →Circles lab →