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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.

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  • $\begingroup$ 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? $\endgroup$ – jmerry Mar 15 '19 at 10:41
  • $\begingroup$ My apologies, I meant it contains one vertex of a square. $\endgroup$ – user354021 Mar 15 '19 at 10:43
  • $\begingroup$ Outer edge of circle points. What's the name for it? $\endgroup$ – user354021 Mar 15 '19 at 10:45
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Here's the situation:

enter image description here

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 $\sqrt 2$.

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enter image description here

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}

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