Where I live there are bike racks that look like circles of metal embedded in the ground. If the rack has a height of $h$ and a base length of $b$, find the radius, $r$, of the circle. Show hint

The centre lies on the perpendicular bisector of the chord (the base of the bike rack), and the distance from the centre to the edge is always $r$.

The base is a chord of the circle, so its perpendicular bisector passes through the centre. We know the distance from the chord to the other side is $h$, so the distance from the chord to the centre is $h - r$. We now have a right-angled triangle with hypotenuse $r$, and sides $h-r$ and $b/2$ (because we bisected the length $b$). Hence, \begin{align*} r^2 &= (h-r)^2 + \left(\frac{b}{2}\right)^2\\ &= h^2 - 2hr + r^2 + \frac{b^2}{4}\\ 2hr &= h^2 + \frac{b^2}{4}\\ r &= \frac{h^2 + b^2/4}{2h}\\ &= \frac{h}{2} + \frac{b^2}{8h}. \end{align*}