The Dome Surface Area Calculator calculates the curved surface area of a dome, bowl or spherical cap — the part of a sphere cut off by a flat plane.
If you know the dome’s base radius and height, the curved area is A = π(a² + h²). If you know the radius of the sphere it comes from, A = 2πRh. The calculator also gives the flat base area and the missing radius, so you can add the base if your dome is closed.
Dome Surface Area Calculator (Spherical Cap)
Curved surface area of a dome, bowl or spherical cap from sphere radius and height, or base radius and height.
Formula
From base radius and height: A = π(a² + h²)
From sphere radius and height: A = 2πRh
Relationships between the measurements:
- R = (a² + h²) ÷ (2h)
- a = √(h(2R − h))
- Base area = πa²
Where:
- A = curved surface area of the dome
- a = radius of the dome’s circular base
- h = dome height (rise from the base to the top)
- R = radius of the full sphere
How the Calculation Works
Step 1: Work out whichever radius you did not enter, using R = (a² + h²) ÷ 2h or a = √(h(2R − h)).
Step 2: Calculate the curved area. Both formulas give the same result, because 2πRh = π(a² + h²).
Step 3: Calculate the base area πa² separately, so you can decide whether to include it.
Variables Explained
| Variable | Meaning | Unit |
|---|---|---|
| a | Base radius (half the span) | cm, m, in, ft |
| h | Dome height | cm, m, in, ft |
| R | Sphere radius | cm, m, in, ft |
| A | Curved surface area | cm², m², in², ft² |
Example Calculation
From base radius and height
Given: base radius = 4 cm, height = 2 cm
Calculation: A = 3.14159 × (4² + 2²) = 3.14159 × 20
Result: A ≈ 62.83 cm² (base area ≈ 50.27 cm², sphere radius 5 cm)
From sphere radius and height
Given: sphere radius = 5 cm, height = 2 cm
Calculation: A = 2 × 3.14159 × 5 × 2
Result: A ≈ 62.83 cm² (base radius 4 cm)
Dome roof in feet
Given: a dome spanning 30 ft (base radius 15 ft) with a 10 ft rise
Result: A = π × (225 + 100) ≈ 1,021.0 ft²
Common Area Conversion Table
| Unit | Equivalent |
|---|---|
| 1 m² | 10.7639 ft² |
| 1 ft² | 144 in² |
| 1 m² | 10,000 cm² |
Shape Properties
A spherical cap is the smaller or larger piece of a sphere cut by a plane. When the height equals the base radius (h = a = R), the cap is exactly a hemisphere and its curved area is 2πR². The curved area depends only on the sphere radius and the height — a result known since Archimedes.
Dimensions Explained
Span, rise and radius
Builders often describe a dome by its span (the base diameter) and rise (the height). Divide the span by 2 to get the base radius a.
Non-spherical domes
These formulas assume the dome is part of a true sphere. Onion, elliptical or pointed domes have different surface areas.
Frequently Asked Questions
How do you calculate the surface area of a dome?
Use A = π(a² + h²) with the base radius and height, or A = 2πRh with the sphere radius and height.
What is the surface area of a spherical cap?
The curved area is 2πRh. Add πa² if you also need the flat base.
How do you find the radius of a dome from its span and height?
R = (a² + h²) ÷ (2h), where a is half the span.
Is a hemisphere a spherical cap?
Yes — it is the special case where the height equals the sphere radius.
How much paint do I need for a dome roof?
Use the curved surface area, then divide by the paint’s coverage per coat from the product label.
