<Dd> π = C d (\ displaystyle \ pi = (\ frac (C) (d))) </Dd> <P> The ratio C / d is constant, regardless of the circle's size . For example, if a circle has twice the diameter of another circle it will also have twice the circumference, preserving the ratio C / d . This definition of π implicitly makes use of flat (Euclidean) geometry; although the notion of a circle can be extended to any curved (non-Euclidean) geometry, these new circles will no longer satisfy the formula π = C / d . </P> <P> Here, the circumference of a circle is the arc length around the perimeter of the circle, a quantity which can be formally defined independently of geometry using limits, a concept in calculus . For example, one may compute directly the arc length of the top half of the unit circle given in Cartesian coordinates by x + y = 1, as the integral: </P> <Dl> <Dd> π = ∫ − 1 1 d x 1 − x 2 . (\ displaystyle \ pi = \ int _ (- 1) ^ (1) (\ frac (dx) (\ sqrt (1 - x ^ (2)))).) </Dd> </Dl>

Who made a contribution to the irrationality of the value of pi