Inscribed circle and square, find the equation for the perimeter of the nth shape in terms of n.

Question

Image is of a square with side 1 which is inscribed in a circle which is inscribed in a square and so on and so forth for ever. How do I find the equation of the nth outer shape in terms of n?

Answer 1

The radius of the first circle is easily found with Pythagore (half of the square's diagonal). This radius is also the half of the length of the second square, thus. You find the radius of the second circle with Pythagore again, and so on: $$c_1 = 1$$ $$r_1 = \frac{1}{2}\sqrt{c_1^2 + c_1^2} = \frac{\sqrt{2}}{2}c_1$$ $$c_2 = 2.r_1 = 2.\frac{\sqrt{2}}{2}c_1=\sqrt{2}c_1$$ $$r_2 = \frac{1}{2}\sqrt{c_2^2 + c_2^2} = \frac{\sqrt{2}}{2}c_2 = \frac{2}{2}c_1 = c_1$$ $$c_3 = \ ...$$