We have investigated the effects of the tri-gate channel structure on electrical properties of extremely thin-body (ETB) InAs-on-insulator $(-{\rm OI})$ MOSFETs. It was found that the tri-gate structure provides significant improvement in short channel effect (SCE) control even in ETB-OI MOSFETs by the simulation. We have fabricated and demonstrated tri-gate InAs-OI MOSFETs with fin width of the top surface down to 40 nm. The tri-gate ETB InAs-OI MOSFETs shows better SCEs control with small effective mobility $(\mu_{\rm eff})$ reduction. Thus, we have demonstrated the operation of sub-20-nm-channel length $(L_{\rm ch})$ InAs-OI MOSFETs. The 20-nm- $L_{\rm ch}$ InAs-OI MOSFETs show good electrostatic with subthreshold slope of 120 mV/decade and drain induced barrier lowering of 110 mV/V, and high transconductance $(G_{m})$ of 1.64 ${\rm mS}/\mu{\rm m}$ . Furthermore, we have realized a wide-range threshold voltage $(V_{\rm th})$ tunability in tri-gate InAs-OI MOSFETs through back bias voltage $(V_{\rm B})$ control.