Let the radius of the circle be r.
Circumference of circle = \\( 2 \\pi r\\)
Area of circle =\\( \\mathrm\\pi r^{2}\\)
Given that, the circumference of the circle and the area of the circle equal.
This implies \\( 2 \\pi r\\) = \\( \\mathrm\\pi r^{2}\\)
r=2
Therefore, the radius of the circle is 2 units.
Hence, the correct answer is A.