How this instrument works
A hemisphere is exactly half of a sphere, produced by slicing straight through the sphere's own center with a flat plane. Because that slice divides the sphere into two identical, mirror-image halves, a hemisphere's volume is simply half of a full sphere's volume: V = (2 ⁄ 3)πr³, exactly half of the familiar (4 ⁄ 3)πr³ formula for a complete sphere.
Like a full sphere, a hemisphere's volume scales with the CUBE of its radius, not with the radius directly — doubling the radius doesn't merely double the volume, it multiplies it by eight (2³). This cubic growth is easy to underestimate: a hemisphere twice as wide looks only modestly larger by eye, but holds eight times as much volume, a distinction that matters directly for anything sized by capacity, from a mixing bowl to a storage tank.
Hemispherical shapes appear often in engineering and architecture specifically because they enclose the maximum volume for a given amount of surface material among all possible dome shapes — the same efficiency that makes a full sphere the most volume-efficient enclosed shape overall, halved. Domes, bowl-shaped tanks, and some pressure-vessel end caps are all built as hemispheres for exactly this reason.
- Enter the hemisphere's radius into the Radius field.
- Read Volume: the sheet applies (2 ⁄ 3)πr³ directly.
- For the equivalent full sphere's volume, simply double the result — the relationship is always exact.
Worked example — a hemisphere of radius 5
A hemisphere has a radius of 5 units. Its volume is V = (2 ⁄ 3) × π × 5³ = (2 ⁄ 3) × π × 125 ≈ 261.80 cubic units — exactly half of the roughly 523.60 cubic units a full sphere of the same radius would hold.
Compare a smaller hemisphere with radius 3: V = (2 ⁄ 3) × π × 27 ≈ 56.55 cubic units. Even though the radius only shrank from 5 to 3, a drop of less than half, the volume fell to roughly a fifth of the original — the cube in the formula amplifying any change in radius far more than a direct, linear relationship would.
Questions
What is the formula for the volume of a hemisphere?
V = (2 ⁄ 3)πr³, where r is the radius. This is exactly half of a full sphere's volume formula, (4 ⁄ 3)πr³, since a hemisphere is simply a sphere sliced precisely through its own center.
How is hemisphere volume different from hemisphere surface area?
They're separate figures for separate purposes: volume (this page) measures the space enclosed inside, while surface area measures the total exterior — the curved dome plus the flat circular base — needed for coating, painting, or material estimates. The two use entirely different formulas.
Does doubling the radius double the volume?
No — volume scales with the CUBE of the radius, so doubling the radius multiplies the volume by 2³ = 8, not by 2. This cubic growth applies identically to a hemisphere and to a full sphere, since a hemisphere's volume is always a fixed fraction of the matching sphere's.
Why are hemispheres common in tank and dome design?
Because a sphere (and therefore a hemisphere, exactly half of one) encloses the most volume for a given amount of surface material among all possible closed shapes — a genuine efficiency advantage that makes hemispherical domes and bowl-shaped tanks a practical choice wherever minimizing material for a given capacity matters.
What is the volume of a hemisphere with radius 0?
Exactly 0 — the hemisphere has collapsed to a single point with no space enclosed at all, the degenerate limit the formula handles cleanly without any special case needed.