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
Earth's circumference system of latitude and longitude. Longditude lines, also known as meridian lines, follow the polar circumference, and are all exactly the same length. Latitude lines, known as circles of latitude are parallel to each other; the equatorial circumference is the largest circle of latitude, and the Earth’s longest circumference distance. Other circles of latitude have smaller circumference; for example, the 60th parallel north or south is approximately half as long as the Equator. According to Cleomedes' "On the Circular Motions of the Celestial Bodies", around 240 BC, Eratosthenes, the librarian of the Library of Alexandria, calculated the circumference of the