Spherical Cap Volume Calculator

What this calculator measures

A spherical cap is the rounded piece you get when a plane slices through a sphere and you keep the top portion. If you picture a dome roof, the end of a pressure vessel, the top of a ball shaved by a flat cut, or a droplet resting on a surface, you are already picturing spherical-cap geometry. The shape is familiar, but its numbers are easy to misjudge by eye. A cap that looks shallow can still have a wide base, and a modest increase in height can add a surprising amount of volume because the cap widens as it grows.

This calculator turns that geometry into four practical outputs: base radius, base area, curved surface area, and volume. Those are the quantities people usually need when estimating storage space, coating area, material coverage, or the footprint of a dome-like feature. Instead of forcing you to derive each expression by hand every time, the page lets you enter the sphere radius and the cap height once, then reports the connected measurements together so you can compare designs or check drawings quickly.

The key idea is that the calculator works from the original sphere, not from the cap alone. You enter the radius of the full sphere, written as R, and the height of the cap, written as h. If your inputs use meters, every length result stays in meters, areas come out in square meters, and volume comes out in cubic meters. The particular unit does not matter; consistency does.

Choosing the right inputs

The sphere radius R is the distance from the center of the sphere to its surface. If a specification gives the sphere diameter instead, divide by two before entering it. The cap height h is measured straight down from the top of the sphere to the cutting plane that forms the cap. This is not the slanted distance along the curved surface. It is a direct depth measurement between the top point and the flat plane.

That distinction matters because cap height controls every other result. When h is tiny, the cap is a thin shallow dome. When h = R, the cap is exactly a hemisphere. When h is larger than R but still no more than 2R, the cap is bigger than a hemisphere and the cutting plane sits below the sphere's center. The calculator accepts all of those cases because they are valid spherical caps. The only impossible input is a height greater than twice the radius, which would imply more than the whole sphere.

If you are measuring a real object, use the inside radius and inside cap height when you want internal capacity, and the outside dimensions when you want outer coating area or external appearance. Mixing inside and outside measurements is a common source of error. Another common mistake is to measure the rise of a dome from a floor or support ring that is not the actual cut plane. When in doubt, sketch the full sphere, mark the top point, draw the plane that creates the cap, and measure h between those two references only.

For quick checking, keep three milestone cases in mind. If h approaches 0, volume should approach 0. If h = R, the cap should behave like a hemisphere. If h approaches 2R, the cap should approach the entire sphere. Those checkpoints make it much easier to spot an accidental unit mix-up or a radius-versus-diameter mistake before you rely on the result.

How the geometry works

The first derived quantity is the base radius, often written as a. It is the radius of the circular rim where the cutting plane meets the sphere. In a cross-section of the sphere, that rim comes from a right triangle, which leads to a compact and useful formula. Once you know a, the base area is simply the area of a circle with radius a.

That expression explains an important visual fact: a shallow cap can still have a fairly large base. The base radius depends on both R and h, and the square-root relationship softens the way it changes. If you are checking a drawing, this output is useful because it tells you the diameter of the circular opening or footprint at the cut plane.

The curved surface area uses an especially elegant result. For a spherical cap, the curved area depends only on the sphere radius and the cap height. You do not need to compute the base radius separately first.

Volume is the quantity most people come for, and it grows nonlinearly with h. That nonlinearity is why guessing by eye is unreliable. Near the top of the sphere, adding a little height barely changes volume because the cap is still narrow. Deeper down, the base widens and the same extra height encloses much more space.

Under the hood, this page is still a calculator in the abstract sense: outputs are functions of inputs. That broader perspective is helpful when you compare multiple scenarios or want to understand how sensitive the result is to one measurement.

Those generic forms do not replace the cap formulas above; they simply frame the same idea more broadly. Here, your two main inputs are sphere radius and cap height, and the displayed results are direct functions of them. If you change only h and keep R fixed, you can immediately see how sensitive base radius, area, and volume are to that one dimension.

Worked example: a dome cut from a sphere of radius 6

Suppose a dome comes from a sphere with radius 6 units and the cap height is 2 units. That is a realistic shallow-dome case: the cap is prominent enough to matter, but it is still much less than a hemisphere. Enter R = 6 and h = 2 in the form, and the calculator computes the following values.

First, the base radius is √(2Rh − h²) = √(24 − 4) = √20 ≈ 4.4721 units. That means the circular opening at the cut plane has a diameter of about 8.9442 units. Next, the base area is πa² = 20π ≈ 62.8319 square units. The curved surface area is 2πRh = 24π ≈ 75.3982 square units. Finally, the enclosed volume is πh²(3R − h)/3 = 64π/3 ≈ 67.0206 cubic units.

These numbers tell a more complete story than volume alone. The base radius shows how wide the cap spreads. The curved area tells you how much exterior finish, paint, membrane, insulation, or panel material might be required on the rounded surface. The volume tells you capacity if the cap forms part of a tank, a lens-like cavity, or a dome-shaped enclosure. Because all four outputs come from the same two measurements, they stay internally consistent and are much less error-prone than mixing formulas from memory.

Notice how the volume does not scale in a simple one-to-one way with height. Doubling h from 1 to 2 does far more than double the volume because the cap is widening at the same time. That is exactly the sort of relationship the calculator makes easy to test.

How to read the result panel

After you press the calculate button, the result area lists the base radius, base area, curved surface area, and volume in that order. Each number uses the same length unit you entered, together with the appropriate square or cubic version for area and volume. If your inputs were in feet, interpret the area outputs as square feet and the volume as cubic feet. If your inputs were in centimeters, the result becomes square centimeters and cubic centimeters.

A quick reasonableness check goes a long way. A cap with a larger height should never have a smaller curved area if the sphere radius stays fixed, because the area formula is linear in h. The volume should always be positive for positive h, and the base radius should never exceed the sphere diameter. If h = R, you should recognize the hemisphere case immediately. If h is very small, expect a tiny volume even when the base diameter still looks respectable.

The copy summary button is helpful when you want to save a scenario, send it to a colleague, or paste it into notes. It captures the main outputs in a sentence so you can compare alternatives later. That is especially useful when you are deciding among multiple dome heights or checking how much capacity changes after a design revision.

Assumptions, edge cases, and common mistakes

This calculator assumes a perfect sphere and a perfectly flat cutting plane. Real parts can deviate from that ideal because of wall thickness, manufacturing tolerances, flattening, or transitions to other shapes. If a tank head blends into a cylinder or includes a knuckle radius, the spherical-cap model is still a useful approximation, but it is not the whole geometry.

The most common mistake is using the wrong reference for h. Remember that cap height is measured from the top of the sphere to the cutting plane, not from the sphere center and not along the curve. The second most common mistake is confusing radius and diameter. A drawing that labels a sphere as 12 units across has radius 6, not 12. The third mistake is mixing units, such as entering radius in meters and height in centimeters. The calculator cannot correct that automatically because the numbers themselves may still look valid.

For practical estimating work, it helps to run two or three nearby scenarios instead of relying on a single exact value. If your measured cap height might be 1.95, 2.00, or 2.05 units, try all three. The spread in the results shows how sensitive your application is to measurement uncertainty. That small habit is often more valuable than carrying extra decimal places, because it reveals whether the geometry is robust or whether a tiny dimensional change causes a meaningful shift in area or volume.

In short, use this page when you know the sphere radius and cap height and you want a fast, reliable geometric summary. The calculator is most useful when paired with a sketch, consistent units, and one good sanity check. Once those are in place, the results are straightforward to interpret and practical to apply.