# Circle containing a point of square and touching two sides

So the problem goes,circle contains one vertex of a square and touches two sides. Length of side is 1cm. What is circumference of a circle? My attempt was trying to find a connection between radius of circle, and side of square, but I'm not sure if Imagined and drew it properly.

• You have not provided enough information for a comprehensible question. Contains one point? Does that mean it's on the circle or inside it? Which point? – jmerry Mar 15 '19 at 10:41
• My apologies, I meant it contains one vertex of a square. – user354021 Mar 15 '19 at 10:43
• Outer edge of circle points. What's the name for it? – user354021 Mar 15 '19 at 10:45

Here's the situation:

The radius from$$(r,r)$$ to $$(1,1)$$ is the hypotenuse of a right triangle of sides $$1-r$$ and $$1-r$$. Express this as an equation, which you can solve for $$r$$. (The equation will have two solutions; just pick the one with $$r<1$$.)

As a short cut, you can just look at that right-angled triangle and note that the ratio of the hypotenuse to the shorter sides is $$\sqrt 2$$.

Extend $$AB$$ one unit to the right to get point $$E$$, $$AD$$ unit upwards to get point $$F$$ and connect these points.

Triangle $$AEF$$ is a right triangle with the sides $$a=b=2$$, $$c=2\sqrt2$$. The circle is the inscribed circle for this triangle.

Recall that for any right triangle with sides $$a,b$$ and hypotenuse $$c$$, the radius of the inscribed circle is just

\begin{align} r&=\tfrac12(a+b-c) ,\\ \text{so }\quad r&=2-\sqrt2 . \end{align}