Area of a circle trigonometry, uses the coarea formula. Define a function formula_10 by formula_11. Note ρ is a Lipschitz function whose gradient is a unit vector formula_12 (almost everywhere). Let "D" be the disc formula_13 in formula_14. We will show that formula_15, where formula_16 is the two-dimensional Lebesgue measure in formula_14. We shall assume that the one-dimensional Hausdorff measure of the circle formula_18 is formula_19, the circumference of the circle of radius "r". (This can be taken as the definition of circumference.) Then, by the coarea formula, Similar to the onion proof outlined above, we could exploit calculus in a different way in
Area of a circle "area" and "circumference". The analytical definitions are seen to be equivalent, if it is agreed that the circumference of the circle is measured as a rectifiable curve by means of the integral The integral appearing on the right is an abelian integral whose value is a half-period of the sine function, equal to . Thus formula_6 is seen to be true as a theorem. Several of the arguments that follow use only concepts from elementary calculus to reproduce the formula formula_1, but in many cases to regard these as actual proofs, they rely implicitly on the fact that one can