Calculate the surface area, volume, base radius, solid angle, and sphere fraction of a spherical cap from sphere radius + cap height or base radius + cap height.
A spherical cap is the portion of a sphere that lies above (or below) a cutting plane. Imagine slicing a ball with a flat cut — the dome-shaped piece you remove is a spherical cap. If the cut passes through the center, the cap is a hemisphere. Otherwise, the cap can range from a thin sliver to nearly the entire sphere.
Given a sphere of radius R and a cap of height h (the perpendicular distance from the flat base to the top of the cap), the lateral (curved) surface area is remarkably simple: A = 2πRh. This formula, discovered by Archimedes, means the area depends only on R and h, not on the base radius. The volume is V = (πh²/3)(3R − h). The base radius a of the cap relates to R and h through a = √(h(2R − h)).
Spherical caps appear throughout science and engineering. Architectural domes — from the Pantheon to modern observatories — are spherical caps. Optical lenses are ground as shallow spherical caps. In geodesy, the "polar cap" of the Earth above a given latitude is a spherical cap. In antenna design, the solid angle subtended by a cap determines the beam coverage. The solid angle Ω = 2π(1 − cos θ) steradians, where θ is the half-angle, measures what fraction of the full sky the cap covers.
This calculator supports two input modes: sphere radius + cap height, or base radius + cap height. It computes all key properties including both surface areas, volume, sphere fractions, solid angle, and half-angle, with presets for common dome, lens, and bowl configurations.
A spherical cap combines several linked measurements that are easy to confuse when worked manually: cap height, sphere radius, base radius, curved area, total area, volume, and angular coverage. In architecture or product design, you may know the dome height and opening width but not the parent sphere. In geometry or physics, you may need the fraction of the sphere or the solid angle rather than just the cap volume.
This calculator is useful because it treats the cap as both a geometric shape and part of a larger sphere. That means you can move from practical measurements to derived values such as half-angle, steradians, and percentage of full-sphere area or volume without rebuilding the setup each time. It is especially helpful for domes, lenses, bowls, lighting coverage, and polar-cap style problems.
Cap surface area: A = 2πRh Cap volume: V = (πh²/3)(3R − h) Base radius: a = √(h(2R − h)) Base area: πa² Sphere from base: R = (a² + h²) / (2h) Solid angle: Ω = 2π(1 − cos θ) sr Half-angle: θ = arccos(1 − h/R) Full sphere area: 4πR² Full sphere volume: (4/3)πR³
Result: Cap surface area ≈ 251.33 cm², cap volume ≈ 435.63 cm³, and base radius = 8 cm.
With mode set to radius-height, sphereR = 10 and capH = 4 define the cap directly. The base radius is √(h(2R − h)) = √(4(20 − 4)) = √64 = 8 cm. The curved cap area is 2πRh = 2π(10)(4) = 80π ≈ 251.33 cm². The volume is (πh²/3)(3R − h) = (π × 16 / 3)(30 − 4) = 416π/3 ≈ 435.63 cm³.
A spherical cap is easiest to understand as a slice of a full sphere cut by a plane. The cap height measures how deep that slice extends from the top of the sphere down to the base plane. As the height changes, the shape can look like a shallow lens, a broad dome, or a full hemisphere. That makes the cap a useful bridge between pure sphere formulas and practical shapes people actually build or measure.
In real projects, the opening size is often what you can measure directly. A dome may be specified by span and rise, a bowl by rim radius and depth, or a lens by aperture and sag. The relationship a = √(h(2R − h)) connects the cap's opening to the parent sphere, which is why switching between radius-height and base-height input modes is so valuable. It lets the same calculator serve both classroom geometry and real fabrication work.
The solid angle output is what makes spherical cap problems especially relevant outside textbook geometry. In lighting, antennas, sensors, and astronomy, you often want to know how much of the full sphere is covered by a cap-shaped region. Steradians provide that answer directly. A cap with a larger half-angle covers more of the surrounding sphere, so the geometry becomes a natural way to describe directional coverage rather than just surface size.
A spherical cap is the region of a sphere above a cutting plane. If the plane passes through the center, the cap is a hemisphere. Otherwise, it is a smaller dome-shaped piece.
The curved (lateral) surface area is A = 2πRh, where R is the sphere radius and h is the cap height. The total area including the base is 2πRh + πa².
V = (πh²/3)(3R − h), where R is the sphere radius and h is the cap height. Use this as a practical reminder before finalizing the result.
R = (a² + h²) / (2h), where a is the base radius and h is the cap height. Keep this note short and outcome-focused for reuse.
A solid angle measures how large an object appears from a point. It is measured in steradians (sr). A full sphere subtends 4π ≈ 12.57 sr. A hemisphere subtends 2π ≈ 6.28 sr.
Architecturally, most domes are spherical caps — portions of a sphere cut by a horizontal plane. However, some domes follow parabolic or catenary curves instead.