The Question
The machine tool diagram shows a symmetric die punch. In this view, the rounded tip is part of a circle of radius r, and the slanted sides are tangent to that circle and form an angle of 54◦. The top and bottom sides of the die punch are horizontal. Use the information in the diagram to find the radius r.
The given diagram
My attempts to solve
ABNM is a rectangle, therefore $$AB = MN$$ $$AC = \frac {AB}{2} = ML$$ $$Θ = \frac {54°}{2} = 27°$$ In △LNK $$ \tan27° = \frac{17/16}{h} $$ $$ h = \frac{17/16}{\tan27°} $$
$$w = h - 1\frac{1}{2}$$ $$\tan27° = \frac{x}{w}$$ $$x = w\tan27°$$
I don't think I've come any close to the solution, so I'll be grateful if you give me some hints or corrections.
You are closer than you think.
Drop a perpendicular from $O$ to $KN$. Since the circle is tangent to $KN$, we have:
$$\frac r{r+w} = \sin 27^\circ$$
or
$$r = \frac {w\sin 27^\circ}{1-\sin 27^\circ}$$