Hemisphere Calculator: Volume, Surface Area & More

The five hemisphere measures

A hemisphere is a sphere (sphere calculator) sliced exactly in half through its center — a curved dome sitting on a flat circular base. Given the radius $r$, every measure follows directly:

$V = \frac{2}{3}\pi r^3 \qquad A = 2\pi r^2 \qquad B = \pi r^2$ $K = 2\pi r^2 + \pi r^2 = 3\pi r^2 \qquad C = 2\pi r$

Volume and curved area are each exactly half of the full sphere's own formulas ($\frac{1}{2}\cdot\frac{4}{3}\pi r^3 = \frac{2}{3}\pi r^3$ and $\frac{1}{2}\cdot 4\pi r^2 = 2\pi r^2$). Total surface area is the one measure that ISN'T simply half of anything — it's the curved dome plus the flat base the cut exposes.

Worked example: radius r = 6

$V = \frac{2}{3}\pi(216) = 144\pi \approx 452.3893 \qquad A = 2\pi(36) = 72\pi \approx 226.1947$ $B = \pi(36) \approx 113.0973 \qquad K = 3\pi(36) = 108\pi \approx 339.2920 \qquad C = 2\pi(6) = 12\pi \approx 37.6991$

Because the radius is exactly 6, volume, curved area, and total surface area all have an EXACT symbolic form ($144\pi$, $72\pi$, $108\pi$) in addition to the decimal approximation. Notice $K = A + B$: $108\pi = 72\pi + 36\pi$ — the total is the curved dome PLUS the flat base, never the curved area alone.

Worked example: radius r = 1

$V = \frac{2}{3}\pi \approx 2.0944 \qquad A = 2\pi \approx 6.2832 \qquad B = \pi \approx 3.1416 \qquad K = 3\pi \approx 9.4248$

A unit hemisphere ($r = 1$) makes every coefficient bare — $\frac{2}{3}\pi$, $2\pi$, $\pi$, $3\pi$ — a clean way to see that $K = A + B$ holds regardless of the radius: $3\pi = 2\pi + \pi$.

Common mistakes

• Using $\frac{4}{3}$ instead of $\frac{2}{3}$ for volume. A hemisphere is HALF a sphere — its volume coefficient is half the sphere's, $\frac{2}{3}$, not the full $\frac{4}{3}$.
• Forgetting the flat base in total surface area. The curved dome alone is only $2\pi r^2$; the total $K = 3\pi r^2$ must add the base disk $\pi r^2$ as well.
• Treating curved area as total area. $A = 2\pi r^2$ (dome only) and $K = 3\pi r^2$ (dome + base) are different measures — a bowl's inner surface is $A$, but the material needed to build a solid dome-on-disk is $K$.

Where this shows up

• Domes, bowls, igloos: the curved surface area tells you how much material covers just the rounded shell (a roof, a mixing bowl's interior).
• The human brain, a planet's hemisphere: volume and total surface area both matter when describing "half" of a roughly spherical object, including the flat cut face.
• Building block for other solids: a hemisphere is exactly half a sphere, and a cylinder with hemispherical caps on each end is a common composite-solid shape (a "capsule").

A hemisphere is defined entirely by its radius — the same one measure that determines a full sphere. Every formula here is either exactly half a sphere's own formula (volume, curved area) or that half plus one new term for the flat base the cut creates (total surface area) — the base area and circumference follow the same rules as any plain circle of that radius.