r2=1.0625
V=21.7
H=4.8
TotalArea[r1_]:=Pi*(r1+r2) *  Sqrt[(r1-r2)^2 +(3(V-Pi r1^2H)/(Pi (r2^2+r1  r2 -2 r1^2)))^2] + Pi( r^2+r1^2) +2Pi r1(H-3(V-Pi r1^2 H)/(Pi(r2^2+r1 r2 -2r1^2)))
Plot[TotalArea[r],{r,1.0625,2}]

Plot[TotalArea'[r],{r,1.1,1.3}]

TotalArea'[r]

FindRoot[TotalArea'[r],{r,1.2}]

R1=1.2197731333239423`
R2=r2

H2=3(V-Pi R1^2 H)/(Pi(R2^2+R1 R2 -2R1^2))
H1 = H-H2