This question emerges from a discussion on quora which concluded that if a circle was drawn on the surface of a sphere, the ratio of radius (from the circle's centre as projected to the sphere's surface, measured over the surface of the sphere) to the circumference could be made to equal exactly 1:3 So there is a "world" in which pi is actual a rational integer.
Q. What is the required ratio of diameter of the sphere to the diameter of the circle for this to happen?
If you take the unit sphere $r=1$, denoted by $S^2$ and take the circle's center to be the north pole $n=(0,0,1)^T$, you want to know the diameter of the circle to be such that $\pi \cdot d = 3$ so $d = \frac3\pi$. From that you can compute backwards the height of the hyperplane $H:= \{x\in\mathbb R^3, x_3 = h\}$ such that $H\cap S^2$ yields this circle of diameter $\frac3\pi$