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
Fife Circle Line line from Inverkeithing that loops back round to Kirkcaldy by an inland route via Cowdenbeath through the old Fife coalfield. Narrowly speaking, just this line could be called the Fife Circle. The current service is actually a combination of two previously separate local routes - Edinburgh to Kirkcaldy & Edinburgh to & . During the 1970s & 80s British Rail only ran a regular daytime service on the Dunfermline line as far as Cowdenbeath; & Cardenden were only served during the weekday business peaks (as can be seen from Table 242 of the UK All Line timetable of that era),