Purpose
I have a basic doubt about the estimation of radius. Let me simulate and find the answer

> r.actual <- 2
> sd <- 1
> n <- 100
> r.realized <- r.actual + rnorm(n, 0, sd)
> estim1 <- sum(r.realized^2)/n - sd^2
> estim2 <- mean(r.realized)^2 - (sd^2)/n
> m <- 1000
> results <- data.frame(estim1 = rep(0, m), estim2 = rep(0, m))
> i <- 1
> for (i in 1:m) {
+     r.realized <- r.actual + rnorm(n, 0, sd)
+     estim1 <- sum(r.realized^2)/n - sd^2
+     estim2 <- mean(r.realized)^2 - (sd^2)/n
+     results[i, 1] <- estim1
+     results[i, 2] <- estim2
+ }
> apply(results, 2, sd)
   estim1    estim2
0.4032214 0.3864239

This is a failure. Instead of using the mean r for calculating area, I deviced a new estimate which was unbiased but not efficient.It is an extremely important learning Basically refer to the theorem of sufficient statistic and how to infer best unbiased estimate from it from casella chapter 7.