# Why squaring RR gives explained variance?

This may be a basic question, but I was wondering why an $R$ value in a regression model can simply be squared to give a figure of explained variance?

I understand that $R$ coefficient can give the strength of a relationship, but I don’t understand how simply squaring this value gives a measure of explained variance.

Any easy explanation of this?

Thanks very much for helping with this!

Hand-wavingly, the correlation $R$ can be thought of as a measure of the angle between two vectors, the dependent vector $Y$ and the independent vector $X$.
If the angle between the vectors is $\theta$, the correlation $R$ is $\cos(\theta)$.
The part of $Y$ that is explained by $X$ is of length $||Y||\cos(\theta)$ and is parallel to $X$ (or the projection of $Y$ on $X$). The part that is not explained is of length $||Y||\sin(\theta)$ and is orthogonal to $X$. In terms of variances, we have
where the first term on the right is the explained variance and the second the unexplained variance. The fraction that is explained is thus $R^2$, not $R$.