Estimate mean and SD of polygon area in Stan

Question

I am trying to calculate the mean and standard deviation of the area of a polygon of which I have sample (x,y) coordinates of the polygon's coordinates (but I don't know the real (x,y) coordinates of the polygon's corners).

This is my model, but I am not sure I am doing this write (it compiles and runs, but I think I am not getting values that make sense.

data {
  int<lower=0> N;            // Number of data points
  int<lower=0> C;
  matrix[C, N] x;               // x coordinates
  matrix[C, N] y;               // y coordinates
}

parameters {
  real<lower=0> mean_a;
  real<lower=0> sd_a;
}

model {
  real area = 0.5 * abs(sum(x[1:N-1,] .* y[2:N,] - y[1:N-1,] .* x[2:N,]));

  // Prior for mean_a and sd_a
  mean_a ~ normal(0, 50);
  sd_a ~ cauchy(0, 20);
  
  // Likelihood
  target += normal_lpdf(area | mean_a, sd_a);
}

and this is an example data set:

{
  "N": 3,
  "C": 4,
  "x": [
    [1, 2, 1],
    [4, 4.5, 4.7],
    [2, 1.5, 1.5],
    [3.8, 4.2, 4]
  ],
  "y": [
    [1, 2, 1],
    [4, 4.5, 4.7],
    [2, 1.5, 1.5],
    [3.8, 4.2, 4]
  ]
}

chatGPT helped with the model specification but it doesn't look right...