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
Mohr's circle points formula_117 on the Mohr circle and the centre of the circle. This double angle relation comes from the fact that the parametric equations for the Mohr circle are a function of formula_35. It can also be seen that the planes formula_80 and formula_81 in the material element around formula_7 of Figure 5 are separated by an angle formula_135, which in the Mohr circle is represented by a formula_136 angle (double the angle). The second approach involves the determination of a point on the Mohr circle called the "pole" or the "origin of planes". Any straight line drawn from the