Another Pi

A circle is a set of points with a fixed distance, so the formula for the circle of radius $r$ centered at the orgin in taxicab geometry is $|x|+|y|=r$ in Cartesian coordianates whose graph is a square, as shown in the picture. The side length of the square equals Manhattan distance between $(r,0)$ and $(0,r)$ , which is $2r$ . Thus, the circle's circumference is $8r$ , making the value of Pi in taxicab geometry $\frac {8r}{2r}=\boxed 4$ .

Alternatively, the formula of arc length in Euclidean geometry $s=\int_a^b\sqrt {1+\left(\frac {dy}{dx}\right)^2}dx$ is replaced by $s=\int_a^b\left(1+\left|\frac {dy}{dx}\right|\right)dx$ in taxicab geometry. In this case, the length of side is given by $l=\int_0^r\left(1+\left|\frac {d}{dx}(r-x)\right|\right)dx=2\int_0^rdx=2r.$