3.25: Badge formed from a sector and a kite

In this question all lengths are in centimetres and all angles are in radians.

The diagram shows a badge which consists of a minor sector $OAB$ of the circle with centre $O$ and radius $12$, and a kite $OBCD$, where $OB = OD$ and $CD = CB$. The arc $AB$ has length $27$. The line $OB$ is perpendicular to the line $CB$, and $COA$ is a straight line.