To find the perimeter of the sector, we need to express it in terms of the radius \(r\) and the area \(A\).

The area of a sector is given by \(A = \frac{1}{2} r^2 \theta\), where \(\theta\) is the angle in radians.

Rearranging for \(\theta\), we have:

\(\theta = \frac{2A}{r^2}\)

The perimeter \(P\) of the sector is the sum of the two radii and the arc length:

\(P = r + r + r\theta\)

Substituting \(\theta\) gives:

\(P = 2r + r \left( \frac{2A}{r^2} \right)\)

Simplifying, we get:

\(P = 2r + \frac{2A}{r}\)