Circle Embedded in Square Distances

Question

In the following situation I know how to find the red distance as (diagonal - diamater) / 2 however I'm not sure how to find the yellow and green distances.

Answer 1

they are equal, if red is $r$ then green\yellow is $r/\sqrt{2}$

Answer 2

In the above diagram:

Construct a square, such that the diagonal (as shown by the red line) connects the outside square and the circle in the shortest length...

Now, we can see that the green and yellow lines are equal (sides of a square)... Let the red line $= red$, Therefore the green and yellow line = $\frac{red}{\sqrt2}$

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To answer this in terms of the radius, $r$:

• Side length of square = $2r$
• Diagonal length = $2r\sqrt2$
• red section = $\frac{1}{2}$diagonal - $r$ $= r\sqrt2 - r$

Therefore Green/Yellow section = $\frac{r\sqrt2 - r}{\sqrt2}$

$$=r - \frac{r}{\sqrt2}$$ $$=r - \frac{r\sqrt2}{2}$$