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 equation of a line can be given in vector form: Here is the position of a point on the line, and is a unit vector in the direction of the line. Then as scalar "t" varies, gives the locus of the line. The distance of an arbitrary point to this line is given by This formula can be derived as follows: formula_39 is a vector from to the point on the line. Then formula_40 is the projected length onto the line and so is a vector that is the projection of formula_39 onto the line. Thus is the component of