There are many sites and posts on this site which tell you how to find the incircle of a triangle in various situations, but none have been about how to find the bounding triangle of a circle.
Say I want to find the triangle that encompasses a circle with a radius of 1.6. This triangle should be a right triangle with two 45° angles. How would I find this? I've tried deriving from existing triangle-to-circle methods, but nothing useful has come out of it.
Suppose you have a 45-45-90 triangle whose legs have length $a$. Then its area $A$ is $\frac{a^2}{2}$ and its perimeter $p$ is $a(2+\sqrt{2})$. So the radius of its incircle is $$ \frac{2A}{p}=\frac{a^2}{a(2+\sqrt{2})}=\frac{a}{2 +\sqrt{2}} $$
In your case, this radius is $1.6$, and so $$a = 1.6(2+\sqrt{2})=3.2+1.6\sqrt{2} \approx 5.46$$