How to calculate sample size in example scenario?

I want to calculate how large my sample should be in the following example scenario:

There are 100 bathrooms being renovated. According to the client, they are all the same size so only one drawing is needed for all the bathrooms. There will be n number of bathrooms 3D scanned in advance with a laser scanner for accurate dimensions. From this n number, the average of each dimension is taken to get the dimensions for the main construction drawing. On these dimensions there is a tolerance of 10 mm(-5, +5) in which the bathrooms can always be renovated neatly.

How big must n be to be able to say that at the average of n with -5, +5 mm tolerances, 95% or 99% of the 100 bathrooms are within said tolerances?