A sun-sized star that has converted the hydrogen in its core to helium is called a red giant.

Here's why:

* Hydrogen Fusion: Stars like our Sun generate energy by fusing hydrogen into helium in their core. This process is called nuclear fusion.

* Core Depletion: Once the hydrogen fuel in the core is exhausted, the star can no longer support itself against gravity.

* Expansion: The core contracts and heats up, causing the outer layers of the star to expand dramatically, becoming much larger and cooler. This cooler temperature shifts the star's light towards the red end of the spectrum, hence the name "red giant".

Important Note: While the core of a red giant is no longer fusing hydrogen, it's important to remember that the star is still fusing hydrogen in a shell surrounding the core. This process contributes to the star's overall luminosity and expansion.