Hemisphere Total Surface Area

A solid hemisphere has two surfaces, and this formula counts both. The curved half-sphere contributes $2\pi r^2$, exactly half the sphere's $4\pi r^2$, and the flat disc that closes it off contributes $\pi r^2$. Together that is $A = 3\pi r^2$. Miss the base and you are a full third short.

So the first question to ask is always which surface you actually mean. Roofing a geodesic dome or gilding the outside of a bowl uses the curved $2\pi r^2$ alone, because the flat face is open air or is bolted to something. Chrome-plating a solid half-ball, costing the wrapper on a dome-shaped chocolate, or computing the total heat-transfer area of a hemispherical tank head that sits on a flat plate all need the full $3\pi r^2$. Many textbooks call $2\pi r^2$ the "curved surface area" and $3\pi r^2$ the "total surface area", and marks are lost on that distinction every exam season.

The curved part hides a lovely fact of its own. Archimedes showed that the sphere's surface area equals that of the cylinder wrapped around it, so a hemisphere's dome has exactly the same area as the label on a can of radius $r$ and height $r$. Wrapping paper for a dome and wrapping paper for that can come to the same square metres.