"Harry" wrote
I know that a half-wave dipole in free space has
a feed-point impedance of approximately 73 ohms.

Can anyone tell me **exactly** how this number is calculated.

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There's no such value as 'exact'.

All you have to do is integrate the power flowing outwards from a
dipole at the centre of an arbitrary sphere with a surface area of x
square metres and equate it to the current flowing in the dipole,
taking into account the distribution of current along the dipole, and
you will obtain the radiation resistance referred to its feedpont.
OK?

But in your case, all you can do is just accept the hearsay value of
73 ohms as being good enough.

Actually it depends on the diameter of the dipole relative to its
length and at HF it is a few percent less. Not that anybody ever
notices such minor discrepancies.
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Reg, G4FGQ