Geodesic Dome Calculator

Calculate the geometry of a dome

This calculator models the outside envelope of a dome as a spherical cap. Enter the measurements you actually know: base diameter and dome height, parent-sphere radius and height, or base diameter and parent-sphere radius. The results include the missing dimensions, curved surface area, circular floor area, curved area plus floor, and enclosed volume.

A hemisphere is one special case. Its height equals its base radius, and the sphere radius also equals the base radius. Many real geodesic domes are taller or shallower than a hemisphere, so using the hemisphere formula for every dome can materially overstate or understate area and volume.

Dome area and volume formulas

Let a be the circular base radius, h the dome height, and R the radius of the parent sphere. The three measurements are related by the Pythagorean theorem.

R = (a² + h²) / (2h)
a = √(2Rh − h²)
curved surface area = 2πRh = π(a² + h²)
floor area = πa²
volume = πh(3a² + h²) / 6

The Wolfram MathWorld spherical-cap derivation gives the same radius, area, and volume relationships. NASA's volume reference gives a full sphere as 4πR³/3; a hemisphere is half of that volume.

Worked example: 20-foot hemispherical dome

For a 20 ft base diameter and a 10 ft height, the base radius is 10 ft. The calculated parent-sphere radius is also 10 ft, so the shape is a hemisphere. Its curved surface area is 628.32 sq ft, its circular floor area is 314.16 sq ft, and its enclosed volume is 2,094.40 cu ft.

If the same 20 ft base has a 12.5 ft height, it is a deeper spherical cap rather than a hemisphere. The calculator derives a 10.25 ft parent-sphere radius, approximately 805.03 sq ft of curved area, and approximately 2,986.15 cu ft of enclosed volume.

Hemisphere reference table

The following values assume the dome is exactly one half of a sphere. Use the calculator for any dome whose height differs from half its base diameter.

Geodesic frame versus spherical envelope

A geodesic dome approximates a curved surface with straight struts and triangular faces. This page calculates the smooth spherical envelope, which is useful for early space, floor, and covering estimates. It does not determine the frequency, class, truncation, strut lengths, hub angles, connector allowances, door openings, panel development, or structural capacity of a particular frame.

Those construction details can change the actual material requirement. Chorded triangular panels sit inside the ideal sphere, seams and overlaps add material, and openings subtract area. Use the dome material calculator for a simple waste allowance, and verify final cut lists against the exact dome system or engineered plans.

Which measurement option should I use?

Base diameter and dome height is the most practical option when planning a room, greenhouse, shelter, or event dome. It uniquely defines a spherical cap and lets the calculator derive the parent-sphere radius.

Sphere radius and dome height is useful when the dome is cut from a known sphere. The height must be greater than zero and no more than the sphere diameter.

Base diameter and sphere radius can describe two caps: a shallow cap and a deep cap. Select the intended section. The base radius cannot exceed the sphere radius.

Surface area for coverings

The curved area is the ideal exterior skin before seams, overlaps, openings, thickness, or waste. Do not add the circular floor area when estimating only the roof covering. Add the floor area only when the project also needs a circular floor or closed base, and keep those materials separate because their specifications often differ.

For a framed geodesic dome, flat triangular panels do not have exactly the same total area as the smooth spherical cap. Treat the curved-area result as an early estimate, then use the actual panel schedule for procurement.

Construction and safety limits

This is a geometry calculator, not a structural design tool. Snow, wind, seismic forces, occupancy, foundations, buckling, connection strength, material grades, local building codes, and safety factors are outside its scope. A larger frequency does not by itself prove that a dome is safe. Use an appropriately qualified designer or engineer for a structure that people will occupy or that must resist environmental loads.

Measure all inputs in the same selected unit. The calculator converts the finished geometry to common metric and US customary equivalents, but mixing an unconverted height in meters with a diameter in feet produces a meaningless shape.

Related dome calculators

Use the dome surface area calculator for a focused area estimate, the dome size calculator for quick size comparisons, and the dome material calculator for covering area with a waste allowance.

Frequently asked questions

What is the surface area formula for a dome?

For a spherical cap, curved area is 2πRh or equivalently π(a² + h²). For a hemisphere, h = R = a, so the curved area simplifies to 2πR².

What is the volume formula for a hemispherical dome?

A hemisphere has volume 2πR³/3. A non-hemispherical spherical cap uses πh²(3R − h)/3, equivalent to πh(3a² + h²)/6.

Does diameter alone define every dome?

No. Diameter alone defines a hemisphere only if you also assume its height equals the base radius. A general spherical cap needs another measurement such as height or parent-sphere radius.

Does this calculate geodesic strut lengths?

No. Strut lengths depend on the selected geodesic subdivision, frequency, class, dome truncation, connector details, and construction method.

Should I include the floor in surface area?

Use curved area for the dome skin. Use curved area plus floor only when estimating a fully closed shape that includes the circular base.