Inscribed circle of a triangle

The circle touches all three sides, so the two pieces of it nearest the right angle make a little square with the two legs: from the corner along each leg is exactly r, the radius.

That leaves 27 − r on one leg and 36 − r on the other. Those two pieces reach the far corners — and from a corner the two touching lengths are equal, so together they are exactly the hypotenuse: (27 − r) + (36 − r) = 45.

Rearranged that is 27 + 36 − 45 = 2r, so r = (27 + 36 − 45) ÷ 2 = 9. The circle inside has a radius of 9 cm.