The normal GEOMEAN function is assuming that all the various numbers are equally weighted, but in this case it's not, so how can I calculate a weighted geometric mean?
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I think you can use Assuming values in
confirmed with CTRL+SHIFT+ENTER so that curly braces like { and } appear around the formula if all the values in A2:D2 (the weights) are the same then that formula will give you the same result as
That formula should give you the same results as the calculator here Note: you can remove the SUM function if weights sum to 1 


There's no builtin formula for this, but you can use SUMPRODUCT to assist. See: http://support.microsoft.com/kb/214049 

