Solution:

Explanation:

1. in order for the straight lines to be 90 deg with the circles, they must be radii of circles with same central point

2. the length of an arc is defined as c = r * θ (where r is the radius, and θ is the angle)

3. we define the inner circle with radius r₁ and its arc L₁ = r₁ * θ₁

4. we define the outer circle with radius r₂ and its arc L₂ = r₂ * θ₂

5. Because of (1), θ₁ + θ₂ = 2π

6. To create the shape, L₁ = L₂ = r₂ - r₁

If you start replacing and solving, you will get a 2nd grade quadratic, which has a positive and a negative solution. The positive solution is that magic number.