Spherical Cap Volume Calculator

Spherical cap measurements calculated here

A spherical cap is the rounded portion left when a flat plane cuts a sphere. Dome roofs, vessel ends, trimmed balls, and liquid droplets can all be modeled as spherical caps when their curved surface belongs to a sphere. The shape may look simple, yet its dimensions are difficult to estimate visually: even a low cap can meet the cutting plane in a broad circle, and added height changes both the footprint and the enclosed space.

This spherical cap calculator returns four linked dimensions from the same geometry: base radius, base area, curved surface area, and volume. Those values are useful for estimating a dome footprint, lining or coating a curved surface, and finding capacity in a cap-shaped space. Enter the sphere radius and cap height once instead of separately reconstructing the geometry for each measurement.

The calculation starts with the full sphere rather than treating the cap as an isolated form. Enter the full sphere radius as R and the perpendicular cap height as h. With meters as the inputs, lengths remain in meters, areas are square meters, and volume is cubic meters. Any length unit is valid as long as both inputs use that same unit.

Entering sphere radius and cap height

For a spherical cap, the sphere radius R is the center-to-surface distance of the original sphere. If a drawing supplies the sphere diameter, divide it by two before using this field. The cap height h is the straight vertical distance from the sphere’s top point to the cutting plane. It is not the distance measured along the curved face.

Cap height drives every reported spherical-cap result. A very small h describes a thin dome. At h = R, the cap is a hemisphere. When h is greater than R but no greater than 2R, the cap contains more than half the sphere and the plane lies below the center. A height above twice the radius is impossible because it would extend beyond the whole sphere.

When measuring a real spherical cap for capacity, use interior dimensions; for exterior finish or appearance, use exterior dimensions. Do not combine an inside radius with an outside height. Also make sure the reference plane is the actual plane that cuts the sphere, not a nearby floor, flange, or support ring. A cross-sectional sketch of the full sphere, its top point, and the cut plane is often the fastest way to confirm the correct height.

Three cap-height checkpoints provide useful error checks. As h approaches 0, cap volume approaches 0. At h = R, the result is a hemisphere. As h approaches 2R, the cap becomes the entire sphere. These cases can expose a diameter entered as a radius or inconsistent units before the values are used in a design estimate.

Spherical cap geometry and formulas

Spherical-cap geometry first determines the base radius, conventionally written as a. This is the radius of the circular rim where the cutting plane intersects the sphere. A right triangle in a cross-section through the sphere center gives the base-radius expression, and the base area then follows from the ordinary circle-area formula.

This spherical-cap relationship explains why a shallow-looking piece may still have a wide circular opening. Both R and h affect the base radius, with the square root moderating the change. The reported base radius can be doubled when a drawing needs the opening or footprint diameter at the cutting plane.

The curved surface area of a spherical cap has a particularly direct expression: it uses only the sphere radius and cap height, so the base radius need not be calculated first.

Spherical cap volume is usually the main result, and it does not increase linearly with h. Near the top of the sphere, a small increase in cap height adds little space because the cap is narrow. Further down, the widening cross-section makes an equal increase in height enclose substantially more volume.

All four spherical-cap outputs come directly from R and h. Holding the sphere radius fixed while changing height is a practical way to see which proposed cut depth has the strongest effect on a dome’s footprint, curved material area, or capacity. Check the height carefully in that comparison, because volume responds to height more sharply than a simple proportional estimate would suggest.

Worked spherical cap example: radius 6, height 2

Consider a shallow dome cut from a sphere with radius 6 units and cap height 2 units. Entering R = 6 and h = 2 gives a cap that is clearly smaller than a hemisphere while still having a substantial circular base.

The base radius is √(2Rh − h²) = √(24 − 4) = √20 ≈ 4.4721 units, so the opening diameter is about 8.9442 units. The base area is πa² = 20π ≈ 62.8319 square units. Its curved surface area is 2πRh = 24π ≈ 75.3982 square units, and its volume is πh²(3R − h)/3 = 64π/3 ≈ 67.0206 cubic units.

These spherical-cap measurements serve different purposes. Base radius describes the spread at the flat cut, curved area helps estimate finish or covering for the rounded face, and volume represents capacity for a cap-shaped enclosure or cavity. Because the calculator derives each value from the same radius and height, the values remain geometrically consistent.

The spherical-cap volume in this table does not follow height one-for-one. Increasing h from 1 to 2 more than doubles volume because the cap also becomes wider. Testing nearby heights in the calculator makes this nonlinear change easy to see before committing to a dimension.

Reading spherical cap results

After calculation, the spherical-cap result panel reports base radius, base area, curved surface area, and volume in that order. Length uses the entered unit, while areas use its square and volume uses its cubic form. For example, feet produce square feet for both areas and cubic feet for volume.

Several checks help confirm a spherical-cap result. With a fixed sphere radius, greater cap height must produce greater curved area because curved area is linear in h. Volume is positive for a positive valid height, and the base radius cannot exceed the sphere radius. A height equal to R should produce the hemisphere case; a very small height should produce a small volume even when the base circle looks relatively broad.

The copy summary button records the main spherical-cap outputs in a sentence for notes or comparison. It is useful when evaluating multiple dome heights, sharing a capacity calculation, or preserving the dimensions used for a drawing revision.

Spherical cap assumptions and measurement pitfalls

This spherical-cap calculator assumes a perfect sphere cut by a perfectly flat plane. Physical components may depart from that model through wall thickness, manufacturing variation, flattening, or transitions into another shape. A tank head that blends into a cylinder or includes a knuckle radius may be approximated by a cap, but its complete geometry is not necessarily a single spherical cap.

The most frequent spherical-cap error is measuring h from the wrong reference. Height runs from the top of the sphere to the cutting plane, not from the center and not along the curved surface. Another common error is entering a diameter as a radius: a sphere 12 units across has radius 6. Finally, avoid mixed units such as meters for radius and centimeters for height, since the calculator cannot infer that mismatch from otherwise valid numbers.

For a measured spherical cap, run a few nearby values rather than relying on one apparently exact input. If cap height may be 1.95, 2.00, or 2.05 units, calculate all three. The range of volumes and areas reveals the impact of measurement uncertainty more clearly than adding decimal places to one estimate.

Use this spherical-cap calculator when the original sphere radius and the perpendicular cap height are known and you need a connected geometric summary. A sketch, consistent units, and a checkpoint such as the hemisphere case make the resulting dimensions easier to trust and apply.