MHB What is the Minimal Area of a Right-Angled Triangle with an Inradius of 1 Unit?

[anemone]

What is the minimal area of a right-angled triangle whose inradius is 1 unit?

 

[Theia]

Spoiler: My attempt

The area of a triangle is \(A = sr\), where \(r\) is the inradius (\(r=1\)) and \(s = (a + b + c)/2\) is the semiperimeter. Apparently, the smallest area is obtained when the semiperimeter - or perimeter - is smallest. This happens, when the triangle is isosceles. The right isosceles triangle with incircle radius 1 has side length $$a = (1 + \sqrt{2})\sqrt{2} = 2 + \sqrt{2}.$$ The area of it is $$A = a^2/2 = 2 + \sqrt{2} + 1 = 3 + \sqrt{2}.$$