Spherical Cap Volume Calculator
Enter the base radius and height of a spherical cap — a dome sliced from a sphere — for its volume, curved area and the radius of the sphere it came from, drawn to scale. Click either dimension on the drawing to change it.
Dimensions
Spherical CapResult
Volume of a spherical cap
A spherical cap is the piece of a sphere cut off by a plane — a dome. The two measurements you can take directly, the base radius a and the rise h, fix everything else, including the radius of the parent sphere: R = (a² + h²)⁄2h.
Editing on the drawing
Click the base diameter or the height on the figure and type a new value — the dome redraws to scale, from a shallow lens to more than a hemisphere.
How to find the volume of a spherical cap
A spherical cap is what a plane slices off a sphere — the dome of a building, the dished end of a tank, the curve of a watch glass. You rarely know the sphere it came from; what you can measure is the width across the flat base and the rise from that base to the top. Those two values, the base radius a and the height h, are all the calculator needs — the parent sphere's radius R follows from them.
The formulas
The volume comes from the two direct measurements. The curved surface is 2πRh, which works out to π(a² + h²) — no need to find R first. Adding the flat base gives the total.
Worked example
Take a dome 92 wide across the base (a = 46) with a rise of 26.
The radius of the sphere the cap was cut from.
From the base radius and height directly.
The dome's skin: 2πRh, or π(a² + h²).
What else the calculator returns
You get the curved (dome) area and the flat base area separately as well as the total surface, plus the parent sphere's radius and the base diameter. When the height equals the base radius the cap is exactly a hemisphere; make it taller and the cap becomes more than half the sphere, which the drawing shows honestly.
Where it comes up
Spherical caps describe domes and cupolas, dished and torispherical tank ends, the liquid at the bottom of a spherical vessel, contact lenses and watch glasses, ball segments in machining, and rivet and rounded fastener heads.
Frequently asked questions
How do you find the volume of a spherical cap?
From the base radius a and height h: V = πh(3a² + h²)⁄6. If you know the parent sphere's radius R instead, the equivalent form is V = πh²(3R − h)⁄3.
What if I know the sphere radius R and the cap height h?
The base radius is a = √(h(2R − h)). Enter that with h, or use V = πh²(3R − h)⁄3 directly.
Is a hemisphere a spherical cap?
Yes — the special case where the height equals the base radius (h = a = R). The formulas then reduce to the hemisphere's ⅔πr³ and 3πr².
What is the curved surface area of a cap?
The dome alone is 2πRh, which works out to π(a² + h²) — it depends only on the two measurements you enter. Add the flat base, πa², for the total.
Can a cap be taller than a hemisphere?
Yes. When h is greater than a, the cutting plane sits below the sphere's centre and the cap is more than half the sphere. The same formulas apply for any height up to the full diameter, 2R.