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
Outer Circle railway line Outer Circle railway line The Outer Circle was built in 1891 as a steam-era suburban railway line, in Melbourne, Australia. It covered much of the modern City of Boroondara, including the suburbs of (from north to south) Kew East, Camberwell, Burwood, Ashburton, and Malvern East. At its longest stage, it ran from Fairfield station, on what is today the Hurstbridge line, to Oakleigh station, on the current Pakenham and Cranbourne lines. The Outer Circle railway was first advocated by a group known as the Upper Yarra Railway League in 1867 who suggested that the Gippsland Railway could be brought into