  We present a new formula for pi involving nested radicals with rapid
convergence. This formula is based on the arctangent function identity with
argument $x=\sqrt{2-{{a}_{k-1}}}/{{a}_{k}}$, where \[
{{a}_{k}}=\underbrace{\sqrt{2+\sqrt{2+\sqrt{2+\cdots
+\sqrt{2}}}}}_{k\,\,\text{square}\,\,\text{roots}} \] is a nested radical
consisting of $k$ square roots. The computational test we performed reveals
that the proposed formula for pi provides a significant improvement in accuracy
as the integer $k$ increases.
