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 to the point P. Hence: We can solve this equation for "x", The y coordinate of the point of intersection can be found by substituting this value of "x" into the equation of the original line, Using the equation for finding the distance between 2 points, formula_35, we can deduce that the formula to find the shortest distance between a line and a point is the following: Recalling that "m" = -"a"/"b" and "k" = - "c"/"b" for the line with equation "ax" + "by" + c = 0, a little algebraic simplification reduces this to the standard expression. The