In the context of differential geometry, π is still π even in the context of non-Euclidean manifolds. The distinguishing feature (again, in differential geomet

In the context of differential geometry, π is still π even in the context of non-Euclidean manifolds. The distinguishing feature (again, in differential geometry) of non-Euclidean geometry is that it has non-zero sectional curvature. One way to measure sectional curvature is to measure the circumference of a small circle and check how it deviates from what would be expected of a Euclidean circle. Euclidean circumference: C(x) = 2 π x. Circumference taking sectional curvature into account, for small x: C(x) = 2 π x - (1/3) K π x^3 + O(x^4), where K is sectional curvature. (The negative sign in the second term means circumferences are smaller than Euclidean when curvature is positive, like on a sphere, and are larger than Euclidean when curvature is negative, like on a saddle.) Curvature can then be measured by taking the third derivative of circumference at 0: C'''(0) = -2 π K. π can still be computed by taking the first derivative of circumference at 0: C'(0) = 2 π, independent of curvature. So even non-Euclidean geometry respects the value of π.