Let us take two variables in a data set $x$ and $y$ and values be $x_1...x_n$ similarly for $y$. And we are looking into the correlation of $x$ and $y$.
Coefficient of correlation is
$$\rho = \frac{Cov(x,y)}{S_xS_y}$$
And Coefficient of Determination is
$$R^2 = \frac{\sigma_{mean}^2 - \sigma_{fit}^2}{\sigma_{mean}^2}$$
How is $\rho^2 = R^2$.