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
Circle line (London Underground) the southern portion of the line. Due to conflict between the two companies it was not until October 1884 that the inner circle was completed. The line was electrified in 1905, and in 1933 the companies were amalgamated into the London Passenger Transport Board. In 1949 the Circle line appeared as a separate line for the first time on the Tube map. In 2009 the closed loop around the centre of London on the north side of the River Thames was broken at Edgware Road and extended west to become a spiral to Hammersmith. The signalling system is being upgraded