The lower part of the house (see the Figure) in the circle is a square, and the top is an equilateral triangle. Find a relation between the length of the side of the house and the radius of the circle
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PLACEHOLDER ANSWER - the answer is a relation. Let the side be $$s$$, with the square's base corners at $$\left(\pm\frac{s}{2}, 0\right)$$ and the apex at $$(0, h)$$, where $$h = s\left(1 + \frac{\sqrt{3}}{2}\right)$$. If the centre is $$(0, k)$$, then $$\frac{s^2}{4} + k^2 = (h - k)^2$$, which gives $$k = \frac{\sqrt{3}}{2}s$$. The radius is $$h - k = s$$, so the side of the house equals the radius of the circle.