The answer is centimeters (cm)
Height is 1D dimension so we cannot use cm^2 or cm^3 to measure some distance. With cm^2 we measure surface (because surface is 2D that is why it has this ^2 next to cm) and cm^3 is unit for measuring volume (volume is 3D it has height, width and thickness that is why it has ^3)
cm^4 cannot be the answer since there is no more than 3 dimensions.