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
Distance from a point to a line are vertical lines). Corresponding sides of these triangles are in the same ratio, so: If point "S" has coordinates ("x","m") then |"PS"| = |"y" - "m"| and the distance from "P" to the line is: Since "S" is on the line, we can find the value of m, and finally obtain: A variation of this proof is to place V at P and compute the area of the triangle ∆"UVT" two ways to obtain that formula_17 where D is the altitude of ∆"UVT" drawn to the hypoteneuse of ∆"UVT" from "P". The distance formula can then used to express formula_18,