The area of C is the area A of the region obtained by rotating r about the origin from 0 to 2π radians, or from 0° to 360° (like the second hand of a clock in reverse).
Let θ = the angle between any radius r and the x-axis. If we divide A into many equal sectors, each with arc-length Δθ , each sector has area ΔA = r² Δθ. As we keep dividing A into more and more sections of smaller arc-length, then, as Δθ → 0, the "area" of the sector ΔA → r.
So, we integrate r dθ from 0 to 2π radians:
A = ∫ r dθ
= r² θ / 2 | θ = 0 to 2π
= (2π r² / 2) - 0
= π r²