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
Circumference in terms of these parameters. One such approximation, due to Euler (1773), for the canonical ellipse, is Some lower and upper bounds on the circumference of the canonical ellipse with formula_5 are Here the upper bound formula_9 is the circumference of a circumscribed concentric circle passing through the endpoints of the ellipse's major axis, and the lower bound formula_10 is the perimeter of an inscribed rhombus with vertices at the endpoints of the major and minor axes. The circumference of an ellipse can be expressed exactly in terms of the complete elliptic integral of the second kind. More precisely, we