Square in a quadrant of a circle (Doubt in a solution given)

From the MindYourDecisions video "Challenging problem given to 12 year olds - square in a quadrant" on YouTube:

Original question:-

I have a doubt in the solution given:- Make 3 additional copies and complete the circle. There will be 5 squares in the shape of a +. One of the diameters of the circle will intersect the outer corners of 2 outer squares.

I have made a very rough diagram following what has being said, I get a shape of +. After that I am confused. {The purple, yellow, and orange squares are the ones I got by symmetry}

Let $$s = \text{side of square}$$. That diameter will be the hypotenuse of a right triangle ($$2$$). The legs are $$s$$ and $$3s$$. So, $$10s^2 = 4\to s^2 = 2/5$$

• Your answer of $2/5$ matches the video's answer of $0.4$, so what's your question? If you're wondering if your approach is correct: it is. (And I consider it an improvement over the one shown in the video.) Congratulations to you! (Or, congratulations to the person whose YouTube comment included that solution.)
– Blue
Commented Dec 25, 2021 at 5:58

"...I have made a very rough diagram following what has being said, I get a shape of +. After that I am confused."

$$(3s)^2+s^2=2^2$$
• So every circular sector would have an inscribed square. Do we have a nice formula for the side in terms of $r$ and $\theta$ ( the central angle)? Commented Dec 27, 2021 at 7:18
• @orangeskid , I don't think there is a 'nice' formula. But it can be easily derived using the method described in the video above. The method in my answer (actually it is not mine) can not give a formula for every $\theta$.