A circle is inscribed in a unit square, and a diagonal of the square is drawn. Find the total length of the segments of the diagonal not contained within the circle.
1. **Calculate the length of the diagonal of the unit square:**
   - A unit square has side length 1.
   - By the Pythagorean theorem, the length of the diagonal \(d\) of the square is given by:
     \[
     d = \sqrt{1^2 + 1^2} = \sqrt{2}
     \]

2. **Determine the diameter of the inscribed circle:**
   - The circle is inscribed in the unit square, so its diameter is equal to the side length of the square, which is 1.

3. **Calculate the total length of the segments of the diagonal not contained within the circle:**
   - The diagonal of the square intersects the circle at two points, creating three segments: two segments outside the circle and one segment inside the circle.
   - The length of the segment of the diagonal inside the circle is equal to the diameter of the circle, which is 1.
   - Therefore, the total length of the segments of the diagonal not contained within the circle is:
     \[
     \text{Total length of segments outside the circle} = \text{Length of diagonal} - \text{Diameter of circle} = \sqrt{2} - 1
     \]

The final answer is \( \boxed{ \sqrt{2} - 1 } \)