if a pool ball diameter == 1 , how can i calculate exact length of the triangle including rounded corners

Question

I'm new here and hope it is ok to post such basic questions. Sometimes things seem simple but they do not turn out that way. I'm trying to make a computer game asset of a pool triangle (15 ball) .

All I know is that the ball in-game is exactly 1 unit. But when I create the asset it has to fit snug just like the picture. Is there a mathematical way to calculate the exact inner length of each side AND then curved corners?

Or should I just trial and error :)

Thanks for reading

Answer 1

If you draw the triangle of the centers of the corner balls, it is equilateral with a side of four diameters. The straight sides of the alignment triangle are parallel to these, offset by one radius. The rounded corners are the same radius as the balls and represent $300^\circ$ of the circle because the ball occupies $60^\circ$.