Surface Area Calculator: Formulas for Every Common Shape

Surface area is the number you need whenever something has to be covered, coated, wrapped or cooled. Volume tells you what fits inside. Surface area tells you what it costs to go round the outside, and those two grow at very different speeds.

Surface area and volume pull apart fast

Take a cube with 2 m sides. Its surface area is 6 faces at 4 m² each, so 24 m², and its volume is 8 m³. Now double the side to 4 m. Surface area becomes 6 x 16 = 96 m², four times bigger. Volume becomes 64 m³, eight times bigger. Double it again and the gap widens again.

That is why a large storage tank holds far more per square metre of steel than a small one, and why small animals lose heat faster than large ones. It also means you cannot scale a paint estimate by eye. If you tackle flat rooms as well, our note on working out square footage covers the two-dimensional version of the same job.

The formulas the calculator is applying

Boxes and cylinders

A rectangular box is 2(lw + lh + wh). For a crate 1.2 m long, 0.6 m wide and 0.5 m high: lw = 0.72, lh = 0.60 and wh = 0.30. Those add to 1.62, and doubled that is 3.24 m² of cardboard, ignoring flaps and overlaps.

A closed cylinder is two circles plus a rolled-up rectangle, so 2πr² + 2πrh. The rectangle part is the giveaway: its width is the circumference, 2πr, and its height is the height of the cylinder.

Spheres and cones

A sphere is 4πr², which is exactly four times the area of the circle you would get by slicing it through the middle. A football with a radius of 0.11 m has a surface area of 4π x 0.0121 = 0.152 m². A cone is the base circle plus the curved part, πr² + πrl, where l is the slant height measured along the side, not the vertical height. Mixing those two up is the single most common cone error.

Using the calculator

Choose the shape first, because that decides which boxes appear. Then enter the dimensions, all in the same unit. Mixing centimetres for one edge and metres for another is the fastest way to get an answer that is out by a factor of ten thousand, since areas scale with the square of the length unit.

Decide whether your object is closed or open. A water tank without a lid is one circle short of a full cylinder. A room you are plastering has four walls and a ceiling but no floor. Subtract the faces you are not covering rather than accepting the closed-shape default, or your material estimate will run high. The same discipline applies when you use an area calculator for flat shapes.

Common questions

How much paint does a given surface area need? Most emulsion covers somewhere around 10 to 12 square metres per litre per coat, and the tin states its own figure. Our 3.24 m² crate at 10 m² per litre needs 0.32 litres a coat, so under a litre for two coats with some left over. Bare or porous surfaces drink far more on the first coat.

Does surface area include the base? It does for a closed shape, which is what a calculator assumes by default. For anything sitting on the ground, or open at the top, take the relevant face off yourself.

What units does the answer come in? Square units matching whatever you typed in. Metres in, square metres out. To convert, remember the factor is squared: 1 m² is 10,000 cm², not 100, and 1 square foot is 144 square inches.

What about a hemisphere? A solid hemisphere is half the sphere plus the flat circular face, so 2πr² + πr² = 3πr². A dome with no floor is just the 2πr² part. Worth checking which one the calculator gives you before ordering material.