To be precise, it's explicitly 2pi R^2 ( 1 - cos(r / R) ) , where r is the radius of the circle on the sphere, and R is the radius of the sphere itself.
If r is small compared to R, cos(r / R) = 1 - (r/R)^2 / 2 + O((r/R)^4) so we recover the usual flat circle formula for the area pi r^2 , which is quadratic in r. Only if the r gets comparable to R does the curve flatten.
So a circular area with radius 1000 km would only be 0.2% smaller on Earth than its flat equivalent. Not a useful way to prove the Earth is round.
On a sphere of radius R, the rate of growth of area approaches 0 as the distance from you approaches 𝜋R (half the circumference), and generally decreases past 𝜋R/2 (quarter circumference)
