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enter image description here

A given square is rotated on its center point by 'z' degrees. A new square is added inside this at no angle, whose size is based on the perimeter of the containing square.

Is there a way to calculate my black square's dimensions, given the angle that the blue square was rotated by and blue's dimensions?

user339946
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  • I'm pretty sure you can do this! Give the angle that the blue square is rotated through some name, say $\theta$. Label the angles in the blue triangles you see. Give the length of a side of the blue square a name, say $L$. Also, notice that all those blue triangles are right-angled. Can you take it from there? –  Jun 10 '14 at 07:10
  • I've figured how to calculate all the angles inside the blue, but not sure how to calculate a black side with that (or if its possible!). – user339946 Jun 10 '14 at 07:12
  • You're almost there. Can you find a relationship for the two blue sides of one of the blue triangles? –  Jun 10 '14 at 07:15

2 Answers2

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The answer is surprisingly simple: $$s = \frac{L}{\cos \theta + \sin \theta},$$ for a enclosing square's side length of $L$ and an angle of rotation $\theta$ between $0$ and $\pi/2$ radians. But I will leave it to you to obtain the derivation of this result.

heropup
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$$\sin(z)=\frac{y}{a}$$

$$\cos(z)=\frac{b}{a}$$

$$b+y=x$$

enter image description here

Vikram
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