R-squared in MANOVA

I performed a (k-means) clusteranalysis and want to use the % of explained variance as a measure for the fit of the 2-, 3-, 4- and 5 clustersolutions I generated. Within the MANOVAs that I performed to check within each clustersolution how the clusters differ on each of the 10 inputdimensions, partial eta squared is calculated.

However, I would like to use the overall explained variance. I understand from reading on this topic that eta-squared and R-squared are used as interchangeable concepts. Is that right and is there a way to calculate r-squared/eta-squared as a function of the MANOVA or based on the output of the MANOVA?

I already found a comment that stated:
Multivariate R-squared = 1 - Wilks' Lambda. Does anyone have any idea if this is correct and where to find more information on this?

Thank you very much for your help in advance.