Assault rifles were developed with two ideas: each soldier carries a weapon capable of full auto fire and the maximum effective range is about 500m. First had sub requirements of lightness and control. All could be achieved by using an intermediate size round. The second requirement came about because it was noticed in WWI most soldiers did fire their rifles at targets over about 4-500m. The rifles used in WWI and WWII are accurate at over 1000 m.
The important dimension is not the caliber of the round but the length of the cartridge. Assault rifle ammunition is longer than pistol but shorter than a Lee-Enfield, Mauser, Moisin-Nagant, or M1 Garand round, which are about the length and caliber. Thus the assault rifle has an effective range longer than a pistol but shorter than the other rifles.