The perimeter of sector \(AOC\) is given by the arc length plus the radii, which is \(2r + r\theta\).

The angle \(COB\) is \(\pi - \theta\) because the total angle in a semicircle is \(\pi\) radians.

The perimeter of sector \(BOC\) is \(2r + r(\pi - \theta)\).

According to the problem, the perimeter of \(BOC\) is twice the perimeter of \(AOC\):

\(2r + r(\pi - \theta) = 2(2r + r\theta)\)

Simplifying, we have:

\(2r + r\pi - r\theta = 4r + 2r\theta\)

\(r\pi - r\theta = 2r\theta + 2r\)

\(r\pi = 3r\theta + 2r\)

\(\pi = 3\theta + 2\)

\(\theta = \frac{\pi - 2}{3}\)

Calculating \(\theta\) gives \(\theta \approx 0.38\) to 2 significant figures.