A hemisphere is simply half a sphere, cut through its center — but that halving affects volume and surface area differently, and it introduces a distinction a full sphere doesn't have: the flat circular base. This calculator handles the common forward direction (radius in, everything else out) and the backward direction (volume, curved surface area, or total surface area in, radius out), while keeping curved and total surface area clearly separate.
Why curved and total surface area are different numbers
A full sphere has one surface area figure: 4πr². Slice it in half and you get two surfaces instead of one — the curved dome you kept, and the flat circular cut you exposed. The curved (dome-only) surface area is 2πr², exactly half the full sphere's. But if you need to cover or paint the entire exterior of a solid hemisphere — dome and base both — you need the total surface area, 3πr², which adds the flat base's circular area (πr²) back in. Confusing the two is the single most common hemisphere calculation mistake: a roofer needs curved SA (the base sits on the building), while someone coating a solid hemisphere-shaped object needs total SA.
Why volume is exactly half, but surface area isn't
Volume splits cleanly in half: a hemisphere's V = 2/3πr³ is precisely half of a full sphere's V = 4/3πr³, because volume is just space, and cutting the sphere in half cuts its enclosed space in half too. Curved surface area also halves cleanly (2πr² vs. 4πr²) for the same reason — you're looking at exactly half the original curved skin. But total surface area does not follow that pattern, because the cut itself creates new surface (the flat base) that didn't exist on the original sphere. That's why 3πr² is more than half of 4πr² — the flat base adds back a chunk of area the simple halving would have thrown away.
When you need to solve backward
Most textbook problems give you the radius and ask for volume or surface area. Real-world problems often run the other way: you know a storage tank needs to hold a certain volume and need to know how big to make the dome lid, or you know how much material covers a hemisphere-shaped shell and need to know what radius that implies. Solving backward means inverting the forward formula — r = ∛(3V ÷ 2π) from volume, r = √(curved SA ÷ 2π) from curved surface area, or r = √(total SA ÷ 3π) from total surface area — and this calculator does all three automatically based on what you select as the known measure.