# Find the radius of the incircle

Here's a small mathematical puzzle I came up with recently:

Find the radius of the larger incircle, given that the radius of the smaller incircle is $$3-\sqrt{3}$$. The hexagon is a regular hexagon, with all sides equal

P.S: the drawing was hand drawn and the incircles don't exactly touch all sides of the triangle. Sorry for that :P

2

Easiest derivation:

obviously the ratio of areas of the two triangles is $$2:1$$; the areas are also proportional to incircle radius times triangle circumference. The circumferences are $$2+\sqrt 3$$ and $$3+\sqrt 3$$ in units of the hexagon side so $$R = 2 (2+\sqrt 3) (3-\sqrt 3)^2 : [(3-\sqrt 3)(3+ \sqrt 3)] = 2$$

• clever approach, but how did you get the circumferences? Aug 6, 2020 at 20:35
• @ThomasL The sides of the triangles in question can all be related to either the height or the side of a regular triangle; the easiest way to make that totally obvious is splitting the regular hexagon into six regular triangles. Aug 6, 2020 at 20:43