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Pythagoras?
Across the diagonal of the 10" box there should be 707mm, across the 14" box 752mm, although you'll lose a bit due to the box corners being radiussed and possibly a bit more from the tyre.
The red addition is roughly a rear 700x23C (668mm OD) by way if illustration, but get someone more competent than a retired woodwork teacher to check the maths before spending money!
MrE
@Dammit
Ok, here's a dimensioned drawing of the case which S&S state is 26" by 26" by 10":
Which is something of a mystery as the internal dimensions are greater than 26".
I'm tempted to order one that is 661mm by 661mm by 254mm internal, based on that, and hope that the United baggage person doesn't have a tape-measure that reads in mm.