How this calculation works
Measure cap height from the sphere pole toward the cutting plane. Allow the full 0–2R range, distinguish the spherical patch from its planar closure and retain the hemisphere/full-sphere limits.
V = πh²(3R−h)/3; curved area = 2πRh; cut radius² = h(2R−h)A worked example
For parent radius 5 m and cap height 2 m, the cutting-plane disk radius is 4 m. Volume is 52π/3 ≈ 54.454273 m³, curved area is 20π m² and the cut disk adds 16π m² if closed.
Good to know
One plane-cut cap, not a two-plane spherical segment, shell or fill-level model. Height cannot exceed the parent sphere diameter. A cap can be larger than a hemisphere.
Reference: MathWorld: geometry definitions and formulas (opens in a new tab) ↗
The parent radius is not usually the cut-circle radius
For parent sphere radius 5 m and cap height 2 m, the cut-disk radius is √(2 × (10−2)) = 4 m. Substituting that 4 m into the parent-radius field would describe a different cap. The sketch marks the parent radius, cap height and disk radius separately.
Height is measured from the cap’s pole to the cutting plane. It is not automatically the distance from the sphere center to that plane. In the default case that center-plane offset has magnitude 3 m, while the cap height is 2 m.
The sphere calculator checks the full-height limit. A cap’s parent radius, cut-disk radius and pole-to-plane height are separate quantities.
A curved patch and a closed cap have different areas
The default volume is π × 2² × (15−2)/3 = 52π/3, about 54.454273 m³. Curved spherical area is 2π × 5 × 2 = 20π m². Closing the cut with its 4 m-radius disk adds another 16π m².
The tool shows both the curved patch and the closed-area total. Neither is a material estimate for a shell, because thickness, inner geometry and seams are not supplied. A two-plane spherical segment is also a different geometry from this single plane-cut cap.
Caps can extend beyond a hemisphere
At height R the cap is a hemisphere. At height 2R it becomes the full sphere and the cut disk shrinks to zero. Heights between R and 2R are valid caps larger than a hemisphere, so the tool does not reject them merely for exceeding one radius.
The sphere calculator checks the full-sphere limit. The capsule calculator instead combines a cylinder with two hemispherical ends. Keep the intended shape and boundary explicit: a spherical cap height is not the straight middle length of a capsule or the height of a cylindrical container.
A few questions, answered
Can height exceed the sphere radius?
Yes. A cap larger than a hemisphere has R<h≤2R. Heights above 2R are outside this model.
Is the cutting disk included in curved area?
No. It is shown separately and added only in the closed-area metric.
What happens at zero height?
The empty-cap limiting volume, area and cut radius are zero.
What happens at twice the radius?
The cap is the full sphere and the cut disk radius is zero.