About the surface area calculator
The surface area calculator finds the total surface area of a sphere, cone, cube, cylinder, rectangular box, capsule, spherical cap, conical frustum, ellipsoid or square pyramid. Where the surface has distinct parts, such as the base and curved surface of a cone, each part is listed.
It is handy when you need to know how much paint, wrap or material covers a solid.
Who should use the surface area calculator
- Students of geometry.
- Painters, packagers and fabricators estimating covering material.
- Anyone comparing the surface of shapes with the same volume.
How to use the surface area calculator: what each input means
- Shape
- Choose the solid.
- Dimensions
- Radius, height or edge lengths in your chosen unit.
- Length unit
- mm to miles; area is in that unit squared.
How the surface area calculator works: method and formula
Each shape uses its standard formula. For example, a cone's total surface area is πr² for the base plus πr × the slant height for the curved surface, where the slant height is √(r² + h²). A general ellipsoid has no closed form, so the calculator uses the Knud Thomsen approximation (p ≈ 1.6075, relative error up to about 1.06%) unless two axes are equal, in which case exact formulas are used.
- Sphere
- A = 4πr²
- Cylinder
- A = 2πr(r + h)
- Cone
- A = πr(r + √(r² + h²))
- Box
- A = 2(lw + lh + wh)
How to read your surface area calculator result
Total surface area includes every face, including bases. If you only need to cover the sides (a tube without ends, say), use the listed parts rather than the total. Add allowance for waste when estimating materials.
Surface Area Calculator worked example
A cone with radius 3 cm and height 10 cm, a 3 cm sphere, and a 6 × 4 × 3 cm box.
- Cone: slant height = √(3² + 10²) = 10.4403; area = π × 3 × (3 + 10.4403) = 126.672 cm².
- Sphere: 4π × 3² = 113.097 cm².
- Box: 2 × (6×4 + 6×3 + 4×3) = 108 cm².
The cone's total surface area is about 126.67 cm².
Practical uses of the surface area calculator
- Estimating paint or wrapping.
- Heat-transfer and coating calculations.
- Homework on solids.
Assumptions and limitations of this calculator
- Ideal shapes only; real objects have thickness and rough surfaces.
- General ellipsoid results are an approximation, and the page states when this is used.
- Hollow tubes are not included as a separate shape here; use the cylinder and subtract.