Why are Giant stars found high on the vertical axis despite potentially cool surface temperatures?

Their sheer size makes them extremely luminous

Stars categorized as Giants or Supergiants reside in the upper regions of the H-R diagram because luminosity is directly tied to both temperature and radius, following the relationship $L imes R^2 T^4$. These stars are typically aging stellar objects that have evolved past the Main Sequence phase, usually after exhausting the hydrogen in their cores. Although their surface temperature might have dropped, causing them to shift towards the right side of the horizontal axis (cooler), their physical radii have expanded enormously—often hundreds of times larger than a star like the Sun. This massive increase in surface area overwhelms the effect of lower surface temperature, resulting in a huge net energy output, thus pushing them high up onto the luminosity (vertical) scale.