UPSC 2002: A hollow sphere of radius R, a hollow cube of side R and a thin circular plate of radius R, made up of the same material, are all heated to 20 °C above room temperature. When left to cool in the room, which of them will reach the room temperature first?

For thin hollow objects of the same material kept at the same excess temperature, the initial cooling rate is proportional to surface area; for identical wall thicknesses the mass is proportional to area, so the ratio of (area / mass) decides the fall in temperature. A hollow sphere of radius R has a much larger surface area (4πR²) than either the cube (6R²) or the circular plate (≈2πR²), giving it the highest area-to-mass ratio and therefore the fastest cooling. Multiple standard answer keys list the sphere as the first to reach room temperature. ([sarthaks.com](https://www.sarthaks.com/66050/hollow-sphere-radius-hollow-cube-side-and-thin-circular-plate-radius-made-the-same-material?utm_source=openai))