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 The line through these two points is perpendicular to the original line, so Thus, formula_7 and by squaring this equation we obtain: Now consider, using the above squared equation. But we also have, since ("m", "n") is on "ax" + "by" + "c" = 0. Thus, and we obtain the length of the line segment determined by these two points, This proof is valid only if the line is not horizontal or vertical. Drop a perpendicular from the point "P" with coordinates ("x", "y") to the line with equation "Ax" + "By" + "C" = 0. Label the foot of