Final answer:
The region directly above the Sun's visible surface known for being hotter is called the chromosphere. It's visible during a total solar eclipse and is the layer right above the photosphere.
Step-by-step explanation:
The hotter region directly above the Sun's visible surface is called the chromosphere. The chromosphere is part of the Sun's atmosphere and lies immediately above the photospheric layers. It is mostly transparent to visible radiation and only emits a small amount of light, making it difficult to observe without a total solar eclipse. The term chromosphere originates from the Greek word for "colored sphere", which was inspired by its narrow red visual appearance during a solar eclipse. The increase in temperature from the chromosphere up to the transition region and the corona is significant, with the chromosphere at about 10,000 K and the corona, the outermost layer, reaching temperatures of a million degrees or more.