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 a side, and "h" is the perpendicular height from the opposite vertex. This proof is only valid if the line is neither vertical nor horizontal, that is, we assume that neither "a" nor "b" in the equation of the line is zero. The line with equation "ax" + "by" + "c" = 0 has slope -"a"/"b", so any line perpendicular to it will have slope "b"/"a" (the negative reciprocal). Let ("m", "n") be the point of intersection of the line "ax" + "by" + "c" = 0 and the line perpendicular to it which passes through the point ("x", "y").