Spherical Cap Calculator

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Walk-through

How to Use This Calculator

3 steps

1

Enter the sphere radius and cap height

Type the sphere's radius (R) and the cap's height (h) — the vertical distance from the flat base of the dome to its highest point — then choose a unit (inches, feet, centimeters, or meters). Both values must be positive, and the cap height cannot exceed the sphere's diameter (2R).

2

Read the cap volume

The Volume tab shows the enclosed volume instantly, plus the curved surface area and base radius as supporting figures, and an interpretation line summarizing the result in plain language.

3

Switch tabs for surface area or base radius

Use the Surface Area tab to see just the curved dome area (useful for roofing or coating estimates), or the Base Radius tab for the radius of the flat circle where the cap meets the rest of the sphere. All three tabs recompute from the same two shared inputs.

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Reference

Formula & Methodology

3 formulas

Volume of a spherical cap

V = πh²(3R − h) / 3

R is the sphere's radius and h is the cap height. Units are cubic (e.g., cubic feet, cubic meters). When h = R, this reduces exactly to the hemisphere volume formula, 2/3πR³.

Curved (dome) surface area

Curved SA = 2πRh

The area of just the rounded, domed part of the cap — not the flat circular base. Units are square (e.g., square feet, square meters). This is the figure roofers and dome builders use, since the base typically sits on a structure and isn't exposed.

Base radius

a = √(h(2R − h))

The radius of the flat circle where the cap meets the rest of the sphere — the rim of the dome. Useful for sizing the opening a dome or tank head sits on.

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Glossary

Key Terms Explained

7 terms

Spherical cap The portion of a sphere cut off by a single plane — picture slicing the top off a ball. What remains, the smaller piece, is the spherical cap: a dome-shaped solid with one curved surface and one flat circular base.

Dome A common real-world name for a spherical cap shape — a rounded roof, a tank head, or a snow globe's curved top are all spherical caps. The volume and surface-area formulas apply directly regardless of what the shape is called.

Sphere radius The radius (R) of the full sphere the cap is cut from — not the radius of the cap's own base. Every other measure in this calculator is derived from R together with the cap height.

Cap height The height (h) of the cap — the perpendicular distance from the flat base of the dome to its highest point. When h equals the sphere radius, the cap becomes exactly a hemisphere; when h exceeds 2R, the shape is no longer geometrically valid.

Base radius The radius of the flat circle where the cap meets the rest of the sphere — the rim of the dome. Computed as a = √(h(2R−h)), expressed in the same linear unit as the sphere radius and cap height.

Curved surface The rounded, dome-shaped part of a spherical cap's surface, excluding the flat base. Curved SA = 2πRh, expressed in square units. This is what needs covering or coating for a dome roof, since the base usually rests on a structure.

Partial sphere Another name for a spherical cap: it's a piece of a sphere, rather than the whole solid. A hemisphere (h = R) is the special case where the cap is exactly half the sphere.

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Scenarios

Real-World Examples

3 worked examples

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Standard calculation

Sphere radius = 5 ft, cap height = 2 ft

Sphere radius 5 ft Cap height 2 ft

With a sphere radius of 5 ft and a cap height of 2 ft, the cap volume is π·2²·(15−2)/3 ≈ 54.45 ft³. The curved surface area is about 62.83 ft², and the base radius — the rim where the cap meets the sphere — is exactly 4 ft. This is the calculator's default example, a clean check that the math lines up.

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Storage tank designer

A dished tank head, sphere radius = 6 ft, head depth = 1.5 ft

Sphere radius 6 ft Cap height 1.5 ft

A shallow dished head like this — common on pressure vessels and storage tanks — holds about 31.81 ft³ inside the domed portion alone, with a curved surface area of roughly 56.55 ft². The base radius of 3.35 ft tells the fabricator the diameter of the flat rim the head welds onto, which is exactly the tank shell's diameter.

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Hemisphere special case

Sphere radius = 5 ft, cap height = 5 ft (h = R)

Sphere radius 5 ft Cap height 5 ft

Setting the cap height equal to the sphere radius produces exactly a hemisphere: volume ≈ 261.80 ft³, curved surface area ≈ 157.08 ft², and base radius = 5 ft — identical to what the Hemisphere Calculator would report for a 5 ft radius. No special-case math is needed; the general cap formulas naturally reduce to the hemisphere formulas when h = R.

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Deep Dive

Spherical cap volume and surface area, explained

3 chapters · 4 min read

A spherical cap is what you get when you slice the top off a sphere with a single flat cut — the smaller piece left over is the cap, and it shows up constantly in engineering as dished tank heads, storage silo domes, and architectural roofs. This calculator takes the sphere's radius and the cap's height and returns the volume, curved surface area, and base radius, with the hemisphere as a built-in special case.

Why the cap formula reduces exactly to the hemisphere formula

A hemisphere is simply the spherical cap you get when the cut passes through the sphere's exact center — in other words, when the cap height h equals the sphere radius R. Plugging h = R into the general volume formula V = πh²(3R−h)/3 gives πR²(3R−R)/3 = πR²(2R)/3 = 2πR³/3, which is exactly the familiar hemisphere volume formula. The same substitution works for curved surface area (2πRh becomes 2πR², the hemisphere's dome area) and base radius (√(h(2R−h)) becomes √(R·R) = R, the hemisphere's flat-circle radius). There's no special-casing needed in the math — the hemisphere is just one point along the continuous range of possible cap heights.

Why curved surface area — not total — is usually what matters

Unlike a hemisphere sitting on the ground, a real spherical cap is often welded, bolted, or built onto a larger structure at its flat base: a dished tank head attaches to a cylindrical shell, a dome roof sits on a drum wall. In both cases, the flat base isn't an exposed surface that needs painting, roofing material, or insulation — only the curved dome is. That's why this calculator reports curved surface area (2πRh) rather than a total that adds the flat circle back in. If you do need the base's own area for a different reason (say, sizing the opening it covers), that's simply the area of a circle with the calculator's reported base radius, πa².

The h > 2R boundary and why it matters

A cap height can range from just above 0 (a vanishingly thin dome cap) up to 2R (the full sphere's diameter), at which point the "cap" is the entire sphere and the base radius shrinks to zero — the flat cut has moved to a single point at the antipode. Any cap height beyond 2R is geometrically impossible: you cannot cut a cap taller than the sphere itself. This calculator blocks that input and explains why, since a silent wrong answer would be worse than an explicit boundary message for anyone sizing a real dome or tank head.

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Questions

Frequently Asked Questions

6 questions

What is the formula for the volume of a spherical cap?+

V = πh²(3R−h)/3, where R is the sphere's radius and h is the cap height. For R = 5 and h = 2, that's π·4·13/3 ≈ 54.45 cubic units.

What happens when the cap height equals the sphere radius?+

When h = R, the spherical cap becomes exactly a hemisphere. The general formulas reduce to the hemisphere formulas automatically — no special-case math needed.

Why do tank and dome builders use spherical cap math?+

Dished or domed tank heads, storage silo caps, and architectural domes are all spherical caps. Knowing the volume and curved surface area is essential for capacity planning and material estimates.

What is the formula for a spherical cap's base radius?+

a = √(h(2R−h)), the radius of the flat circle where the cap meets the rest of the sphere. For R = 5 and h = 2, a = √16 = 4.

What is the formula for a spherical cap's curved surface area?+

Curved surface area = 2πRh — the rounded dome surface only, not the flat circular base. For R = 5 and h = 2, that's 20π ≈ 62.83 square units.

What units does the Spherical Cap Calculator use?+

Volume is shown in cubic units (matching your chosen linear unit cubed), while surface area uses square units and base radius uses the same linear unit as the sphere radius and cap height.

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Spherical Cap Calculator

Sphere & Cap Dimensions

How to Use This Calculator

Enter the sphere radius and cap height

Read the cap volume

Switch tabs for surface area or base radius

Formula & Methodology

Key Terms Explained

Real-World Examples

Standard calculation

Storage tank designer

Hemisphere special case

Spherical cap volume and surface area, explained

Why the cap formula reduces exactly to the hemisphere formula

Why curved surface area — not total — is usually what matters

The h > 2R boundary and why it matters

Frequently Asked Questions

Get a branded PDF of your results

Spherical Cap Calculator

Sphere & Cap Dimensions

Related Calculators

How to Use This Calculator

Enter the sphere radius and cap height

Read the cap volume

Switch tabs for surface area or base radius

Formula & Methodology

Key Terms Explained

Real-World Examples

Spherical cap volume and surface area, explained

Why the cap formula reduces exactly to the hemisphere formula

Why curved surface area — not total — is usually what matters

The h > 2R boundary and why it matters

Frequently Asked Questions

Get a branded PDF of your results

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