# If we suppose that a party balloon has a diameter of 16cm, how many balloons can lift a 75 kg man?

~ under standard atmospheric conditions

Relevance

You will have to assume the balloons are spherical (as to be able to calculate their volume) and that they are filled with a lighter gas than air at STP (say Helium).

A spherical balloon of diameter D will have volume 1/6 pi D^3 . N of these will have volume

V = N pi D^3 / 6

The total buoyant force will be

Fb = N pi D^3 / 6 * ( rho_air - rho_Helium) g

You want this to exceed mg, where m is the man's mass:

N pi D^3 / 6 *( rho_air - rho_Helium) g > m g

N > 6m/( pi D^3 ( rho_air - rho_Helium) )

N> 6*75kg / ( 3.1415 * (0.16m)^3 * ( 1.21 kg/m^3 - 0.178 kg/m^3) )

N>33888

That amounts to a total volume of about 72 m^3 of balloons.

To get a mental picture of that, if you would fit all these balloons in a giant spherical net and assume close packing, that net would have a diameter of

d_net = (72m^3 * 6 / Pi )^(1/3) = 5.2 meter .

Very doable!