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Spherical Cap Calculator

Volume equals pi h squared over three times the quantity three R minus h

Cap height must satisfy 0 < h ≤ 2R (h equals 2R for a full sphere; h equals R for a hemisphere).

Solution

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Calculate Spherical Cap Volume

Use this form when the full sphere radius R and the cap height h are known. The cap is the dome-shaped region cut off by a plane; this is the standard partial-fill formula for a spherical tank.

V = (π h² / 3)(3R − h)

Calculate Spherical Cap Curved Surface Area

Use this form for the curved dome surface only (no flat base). The result is linear in h — a half-height cap covers half the curved surface of a full sphere of the same radius.

S_curved = 2π R h

Calculate Spherical Cap Total Surface Area

Use this form for the curved dome PLUS the flat circular base disk. Equals 2πRh + πh(2R − h), with the second term being the base area πa².

S_total = π h (4R − h)

Calculate Sphere Radius from Cap Volume and Height

Inverse form: given the cap volume V and the cap height h, recover the full sphere's radius R. Useful when designing a tank to hit a target partial-fill capacity at a known fill depth.

R = V / (π h²) + h / 3

How It Works

A spherical cap is the dome-shaped region cut from a sphere by a single plane. It is fully characterised by the full sphere radius R and the cap height h (perpendicular distance from the cutting plane to the top of the cap). This calculator solves V = (π h² / 3)(3R − h) for volume, S_curved = 2π R h for the curved dome surface, S_total = π h (4R − h) for curved + flat base, and the inverse R = V/(π h²) + h/3 to back out the sphere radius from a target volume. The base radius a = √(h(2R − h)) and base disk area π a² are always shown as supplementary outputs.

Example Problem

A spherical pressure tank has interior radius R = 5 m and is filled with liquid to a depth of h = 2 m measured up from the bottom of the tank. The liquid forms a spherical cap. Compute the volume of liquid, the wetted curved surface area, the total wetted area, and the radius of the free-surface circle.

  1. Knowns: R = 5 m (full sphere radius), h = 2 m (cap height = liquid depth).
  2. Volume: V = (π · h² / 3)(3R − h) = (π · 4 / 3)(15 − 2) = (4π / 3) · 13 = 52π / 3 ≈ 54.4543 m³.
  3. Curved surface area (wetted dome): S_curved = 2π R h = 2π · 5 · 2 = 20π ≈ 62.832 m².
  4. Base radius (free-surface circle): a = √(h(2R − h)) = √(2 · 8) = √16 = 4 m.
  5. Base area (free-surface disk): π a² = 16π ≈ 50.265 m².
  6. Total surface area (curved + base): S_total = π h (4R − h) = π · 2 · 18 = 36π ≈ 113.097 m². Sanity check: 20π + 16π = 36π ✓.
  7. Inverse check: R = V / (π h²) + h / 3 = (52π/3) / (4π) + 2/3 = 13/3 + 2/3 = 5 m, recovering the input radius.

All four spherical-cap formulas degenerate cleanly: at h = R the cap is a hemisphere (V = (2/3)π R³, S_curved = 2π R²); at h = 2R the cap is the full sphere (V = (4/3)π R³) and the base radius collapses to zero.

When to Use Each Variable

  • Solve for Volume — when R and h are known and you need the dome's volume — partial-fill of a spherical tank, dome enclosure capacity, or volume of a contact lens.
  • Solve for Curved Surface Area — when you need only the dome's outer surface — wetted area in a partially filled sphere, paint or coating for an open-bottom dome.
  • Solve for Total Surface Area — when both the curved dome and the flat base disk matter — closed-bottom dome roofs, hemispherical-with-floor enclosures.
  • Solve for Sphere Radius — when you know the target cap volume and cap height — sizing a spherical tank to hit a partial-fill capacity at a documented liquid depth.

Key Concepts

A spherical cap is one of three closely related sphere sections. (1) When h = R the cap is a hemisphere — half a sphere. (2) When h < R the cap is shallower than a hemisphere (think contact lens or shallow dome). (3) When R < h < 2R the cap is more than half a sphere (a spherical 'bowl with overhang'). The base radius a is the radius of the circle where the cutting plane intersects the sphere — it grows from zero at h = 0, peaks at a = R when h = R (the equator), then shrinks back to zero at h = 2R as the cutting plane reaches the bottom pole. Volume is monotonic in h: more cap height always means more volume, but the relationship is not linear because the cross-section widens then narrows.

Applications

  • Storage and process tanks: liquid volume in a partially filled spherical or hemispherical tank as a function of fill depth
  • Architecture: dome roofs, planetariums, gas-holder caps — volume and surface area for HVAC and structural sizing
  • Optics and biomedical: contact lens curved area (sagittal cap), corneal cap geometry, sclera measurements
  • Astronomy: spherical-cap viewport on an observatory dome, lit-fraction of the moon when modeled as a cap
  • Aerospace: spherical-cap nose cone heat-shield surface area

Common Mistakes

  • Confusing cap height h with sphere radius R — h is the height of the dome above the cutting plane, not the sphere's radius. They are equal only in the hemisphere special case.
  • Forgetting the constraint h ≤ 2R — a cap height greater than the sphere diameter is geometrically impossible.
  • Using V = (4/3)π R³ (full sphere) when the tank is only partially filled — substitute the cap formula V = (π h²/3)(3R − h) instead.
  • Mixing the base radius a with the sphere radius R — a depends on h, R is fixed by the sphere itself.
  • Forgetting whether to add the flat base when computing surface area — closed-bottom domes use S_total = π h (4R − h), open-bottom domes use just the curved S_curved = 2π R h.

Frequently Asked Questions

How do you calculate the volume of a spherical cap?

Use V = (π h² / 3)(3R − h), where R is the full sphere radius and h is the cap height (depth of the dome). For R = 5 m and h = 2 m, V = (4π/3) · 13 = 52π/3 ≈ 54.45 m³.

What is a spherical cap?

A spherical cap is the dome-shaped region of a sphere cut off by a single plane. The cutting plane creates a flat circular base; everything on one side of that plane is the cap. When the plane passes through the sphere's centre, the cap is exactly a hemisphere.

How do you find the volume of liquid in a partially filled spherical tank?

If the tank has interior radius R and the liquid depth (measured from the lowest point of the tank up to the free surface) is h, the liquid volume is V = (π h² / 3)(3R − h). This is the standard partial-fill formula for a spherical tank. For a half-full hemispherical end-cap, h = R and V reduces to (2/3)π R³.

What is the surface area of a spherical dome?

The curved dome surface (no floor) is S_curved = 2π R h — surprisingly simple, and linear in h. If you also need the floor (the flat base disk), add π h (2R − h), giving the total S_total = π h (4R − h). For R = 5 m, h = 2 m: curved 20π ≈ 62.8 m², total 36π ≈ 113.1 m².

What is the formula for the base radius of a spherical cap?

The base radius is a = √(h(2R − h)), where R is the sphere radius and h is the cap height. This is the radius of the circular cross-section where the cutting plane intersects the sphere. At h = R it reaches its maximum a = R (the sphere's equator); at h = 0 or h = 2R it is zero (a single point of tangency).

How is a spherical cap different from a hemisphere?

A hemisphere is the special case of a spherical cap where h = R — the cutting plane passes through the sphere's centre and the cap is exactly half the sphere. A general spherical cap can be shallower (h < R, a shallow dome) or deeper (R < h < 2R, more than half a sphere). All hemisphere formulas are spherical-cap formulas evaluated at h = R.

Can a spherical cap's height exceed the sphere's diameter?

No. The cap height h must satisfy 0 < h ≤ 2R. At h = 2R the 'cap' becomes the whole sphere and the base radius collapses to zero (the cutting plane is tangent at the opposite pole). The calculator rejects h > 2R as geometrically impossible.

How do you find the sphere's radius given the cap volume and height?

Rearrange V = (π h² / 3)(3R − h) for R: R = V / (π h²) + h / 3. For V = 52π/3 m³ and h = 2 m, R = (52π/3) / (4π) + 2/3 = 13/3 + 2/3 = 5 m, recovering the sphere radius.

Reference: Weisstein, Eric W. "Spherical Cap." MathWorld — A Wolfram Web Resource. https://mathworld.wolfram.com/SphericalCap.html

Worked Examples

Partial Tank Fill

How much liquid is in a 5 m spherical tank filled to 2 m depth?

A spherical pressure vessel of radius R = 5 m holds liquid to a depth of h = 2 m. Find the liquid volume — the liquid forms a spherical cap.

  • Knowns: R = 5 m, h = 2 m
  • Formula: V = (π h² / 3)(3R − h)
  • V = (π · 4 / 3)(15 − 2) = (4π / 3) · 13 = 52π / 3 ≈ 54.45 m³

Volume ≈ 54.45 m³ (about 14,380 US gallons)

Even though the tank is only 20% full by depth (2 m of 10 m diameter), it already holds about 10% of the tank's total capacity — sphere cross-sections widen quickly near the bottom.

Dome Roof

How much paint covers a dome roof with 8 m base radius and 3 m height?

A spherical-cap dome roof has a base (footprint) radius a = 8 m and a peak height of h = 3 m above the wall. Compute the curved surface area for paint estimates.

  • Solve for the sphere radius: a² = h(2R − h) ⇒ R = (a² + h²) / (2h) = (64 + 9) / 6 ≈ 12.17 m
  • Confirm h ≤ 2R: 3 ≤ 24.33 ✓
  • Formula: S_curved = 2π R h = 2π · (73/6) · 3 = 73π m²
  • S_curved ≈ 229.34 m²

Curved surface area ≈ 229.3 m² (≈ 2,468 ft²)

Loading this example into the calculator uses R ≈ 12.167 and h = 3 m. To go straight from the footprint radius a to the dome roof's sphere radius, use R = (a² + h²)/(2h).

Contact Lens

What is the curved area of a soft contact lens?

A soft contact lens models as a shallow spherical cap with base curve (sphere radius) R = 8.6 mm and sagitta (cap height) h = 1.5 mm. Compute the curved surface area to estimate oxygen-permeation surface.

  • Knowns: R = 8.6 mm, h = 1.5 mm
  • Formula: S_curved = 2π R h
  • S_curved = 2π · 8.6 · 1.5 = 25.8π mm²
  • S_curved ≈ 81.05 mm² (≈ 0.81 cm²)

Curved (anterior) surface area ≈ 81.05 mm²

Real contact lenses are aspheric and have a thinner edge, but the spherical-cap model captures the dominant area term within a few percent.

Spherical Cap Formulas

A spherical cap is characterised by the full sphere radius R and the cap height h (depth of the dome from cutting plane to top). All four quantities below derive from that pair.

Volume equals pi h squared over three times the quantity three R minus hCurved surface area equals two pi R hTotal surface area equals pi h times the quantity four R minus hSphere radius equals V over pi h squared plus h over three
Spherical cap cut from a sphere of radius R by a horizontal planeRha

Where:

  • V — volume of the spherical cap (the dome region)
  • S_curved — curved dome surface area, excluding the flat base disk
  • S_total — curved dome plus the flat base disk
  • R — full sphere radius (the sphere from which the cap is cut)
  • h — cap height (perpendicular distance from cutting plane to top of cap; 0 < h ≤ 2R)
  • a — base radius, the radius of the circle where the plane cuts the sphere: a = √(h(2R − h))

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