(To answer your question, the unit circle, with radius one, has area π.)
A quarter arc of the unit circle is π/2. A quarter arc of the circle with area 2π, though... Here, let's do it in τ for fun. :3
So, the circle with area τ has radius sqrt(2) (from A = πr^2), diameter 2sqrt(2), and circumference τsqrt(2) (from C = Dπ), thus a quarter arc of that circle is τsqrt(2)/4. That's not a pretty number at all to work with.
Maybe keeping the unit circle at its current size is a better idea.
A quarter arc of a circle is π/2 radians, regardless of area.