Annoying conics are annoying

Geometry Level 2

The smallest possible circle tangent to both sides of a hyperbola \[ x^{2}-y^{2}=1\] circumscribes an ellipse with minor axis half the length of the major axis. The largest possible circle is inscribed in this ellipse. The area of the region within both circles but not containing the area within the ellipse can be expressed as Z pi. Find Z in decimal.

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