I am in no way a mathematician, I am a graphic designer by trade but I do like precision in what I do.
Quite a tricky one to describe hence the attached graphic, however, in words;
I have punched a circle out of a square of the same dimensions. I have discarded all but the top left corner shape. I'd now like to round this corner off. So at 30 degrees from the top right I have created a point, from this point I'd like to draw a circle (green) until it fills the available space.
I therefore want to calculate the radius of the green circle. But my limited maths knowledge means I'm struggling and would be very grateful for any help.
I have the angles a, b and c. I also have the lengths of d, e and f.
If radius of green circle is $r$, then for length $OD$: $$ OD = OB+BC+CD,\\ \frac d{\cos a} = d + r + \frac r{\cos a},\\ r\left(\frac 1{\cos a}+1\right) = d\left(\frac 1{\cos a}-1\right) r = d\frac{1-\cos a}{1+\cos a} $$
If $a=30°$, then $r = (7-4\sqrt 3)d \approx 0.0718\ d$.