Volume of a bead with square hole- Program approach?

[Werner]

And the corresponding 42-style program!
Since the formula has gotten relatively simple, I decided to go for a solver program:

Code:

00 { 106-Byte Prgm }
01▸LBL "RBV"
02 MVAR "D"
03 MVAR "A"
04 MVAR "B"
05 MVAR "V"
06 RCL "D"
07 X↑2
08 LSTO "D2"
09 RCL "A"
10 X↑2
11 -
12 RCL "B"
13 X↑2
14 -
15 SQRT
16 LSTO "K"
17 RCL "A"
18 RCL× "B"
19 STO× ST Y
20 RCL÷ "D"
21 RCL÷ "K"
22 ATAN
23 RCL× "D"
24 RCL× "D2"
25 -
26 3
27 ÷
28 RCL "A"
29 RCL "B"
30 XEQ 10
31 RCL- "V"
32 RCL "B"
33 RCL "A"
34▸LBL 10
35 RCL÷ "K"
36 ATAN
37 X<>Y
38 STO× ST Y
39 X↑2
40 3
41 ÷
42 RCL- "D2"
43 ×
44 2
45 ÷
46 -
47 END

Use with shift-SOLVER, select RBV, enter 3 values and then press the variable you want to solve for. For subsequent runs you need only enter the values that have changed.
For those who insist on typing it all in: do a 0 STO "var" for each variable used by the program (var=a,b,d,d2,k). Then you can select the variable from the menu when using STO etc.

Cheers, Werner