Math Calculators
Surface Area Calculator
Surface area is the area of the outside of a solid. A box has six faces, a closed cylinder two circular ends and a curved side, a sphere its whole skin, a closed cone its base plus the curved surface, and a capsule its two hemispherical ends. Enter the solid and its dimensions.
Calculator
Formula
Box: SA = 2 × (l × w + l × h + w × h), the six faces added in three pairs. Closed cylinder: SA = 2 × π × r × (r + h), the two circular ends plus the curved side. Closed cone: SA = π × r × (r + √(r² + h²)), the circular base plus the lateral surface. Sphere: SA = 4 × π × r², where r is the radius. Capsule: SA = 4 × π × r² + 2 × π × r × h, the two hemispherical ends plus the cylinder between them. The area comes back in the unit you measured in, squared, with square metres (1 ft² = 0.09290304 m²) and square feet beside it.
About This Calculator
What It Does
This calculator works out the area of the outside of a solid. Choose a box (rectangular prism), a closed cylinder or a sphere, enter the dimensions that solid needs in millimetres, centimetres, metres, inches, feet or yards, and it applies the matching formula: 2 × (length × width + length × height + width × height), 2 × π × r × (r + h), π × r × (r + √(r² + h²)) for a closed cone, 4 × π × r², or 4 × π × r² + 2 × π × r × h for a capsule. The surface area comes back in the unit you measured in, squared, with square metres and square feet beside it; a dimension that is zero, negative or missing is rejected and reported.
Worked Example
Example — the page opens on a box of 2 m by 3 m by 4 m: 2 × (2 × 3 + 2 × 4 + 3 × 4) = 2 × 26 = 52 m², which is 559.723 ft². A closed cylinder of 1 m radius and 2 m height is 2 × π × 1 × 3 = 18.8496 m², and a sphere of 2 m radius is 4 × π × 2² = 50.2655 m². A closed cone of 1 m radius and 2 m height is π × 1 × (1 + √5) = 10.1664 m², and a capsule of 1 m radius with a 2 m cylindrical body is 4 × π × 1² + 2 × π × 1 × 2 = 25.1327 m².
Real-World Usage
Surface area prices and orders the material a solid's skin takes: paint and varnish for a crate, a wrap or a label for a tin, insulation or cladding for a tank, fabric for a dome. It is also the figure that decides heat loss, drying time and coating weight, all of which scale with the outside rather than the volume — doubling every side multiplies the surface by four and the volume by eight.
Local Context
Both systems are shown because the page serves every country at once: metric readers take the square metres, US and UK readers the square feet (1 ft² = 0.09290304 m²). For a cylinder the page always reports the full outside — both ends and the curved side — so sheet metal, paint or a label is never ordered short by a lid nobody mentioned.
Last reviewed: October 2026.
Frequently Asked Questions
Does the cylinder include the top and the bottom?
Yes — the cylinder here is closed, so the answer covers both circular ends and the curved side: 2 × π × r × (r + h). A cylinder of 1 m radius and 2 m height is 18.8496 m². If the top is open, subtract the lid, π × r² = 3.14159 m², leaving 15.708 m²; a pipe with both ends open keeps only the curved side, 2 × π × r × h = 12.5664 m².
What is a surface area used for?
It is the amount of material a solid's outside takes: paint or varnish for a crate, a label or a coating for a tin, fabric for a dome, sheet metal to clad a tank. A box's six faces come in three matching pairs, which is why the formula is 2 × (length × width + length × height + width × height) rather than a sum of six separate terms.
Why is the result in square metres while a sphere asks only for a radius?
A surface is a flat area, so its unit is a length squared, exactly as on the area calculator: 2 m × 3 m is 6 m². A sphere is fixed by its radius alone — 4 × π × r², so a 2 m radius gives 50.2655 m² — because every point on a sphere lies the same distance from the centre, so there is no second measurement to give. The metric and imperial counterparts sit beside the result: 1 ft² = 0.09290304 m².
Why is a zero or a negative measurement refused?
A side of length zero is not a solid and a negative one has no geometric meaning, so the field is rejected and the result stays at 0 with the unit named, rather than showing a plausible-looking surface for a shape that cannot exist. The same guard covers all three surfaces in this family, so an impossible entry never reaches the arithmetic.