How to calculate the area of ellipse drawn by ggplot2?

In ggplot2, after I drawing the ellipse plot using stat_ellipse, is there any way to calculate the area of this ellipse? Here is the code and the plot:

library(ggplot2)
set.seed(1234)
x <- rnorm (1:1000)
y <- rnorm (1:1000)
data <- cbind(x, y)
data <- as.data.frame(data)
ggplot (data, aes (x = x, y = y))+
  geom_point()+
  stat_ellipse()

You can calculate the area of the ellipse by finding its semi-major and semi-minor axes (as shown in this SO answer):

# Plot object
p = ggplot (data, aes (x = x, y = y))+
  geom_point()+
  stat_ellipse(segments=201) # Default is 51. We use a finer grid for more accurate area.

# Get ellipse coordinates from plot
pb = ggplot_build(p)
el = pb$data[[2]][c("x","y")]

# Center of ellipse
ctr = MASS::cov.trob(el)$center  # Per @Roland's comment

# Calculate distance to center from each point on the ellipse
dist2center <- sqrt(rowSums((t(t(el)-ctr))^2))

# Calculate area of ellipse from semi-major and semi-minor axes. 
# These are, respectively, the largest and smallest values of dist2center. 
pi*min(dist2center)*max(dist2center)

[1] 13.82067