Does it mean we can never know the exact area of circle?

Question

I performed three calculations for a circle of radius $10^7$ metre

$$(\pi-3.14)(10^7)^{2}=1.59265359\times10^{11}$$

$$(\pi-3.1415926535897)(10^{7})^2=9.31$$

$$(\pi-3.141592653589793238462)(10^{7})^2=0.00000006433832795028842$$ It implies large difference in the calculated area of the circle and it's dependence on the accuracy of value of $\pi$ used.

Since we may never know the actual value of $\pi$,Does it imply that it is impossible to calculate the exact value of the area of a circle?

Answer 1

We have

$S=\pi R^2$ assume $R$ is a constant.

take ligarithmic derivative,

$\frac{\delta S}{S}=\frac{\delta\pi}{\pi}$

$\delta S=\delta \pi.R^2$

so if $R$ is large, the error on the surface is also large.