Surface Area Calculator
Pick a solid — cube, sphere, cylinder, cone or square pyramid — enter its dimensions and get the total surface area, plus the lateral and base components, in any unit.
Total surface area (SA = 6a²)
150 square units
- Lateral surface area
- 100
- Base area (per face)
- 25
Surface area is the sum of all outer faces of the solid. The lateral component is everything except the flat bases; the base area is one flat face (the cylinder has two; the pyramid one). All formulas use exact π — rounding happens only at display time.
How to use this calculator
Choose the shape you want from the dropdown. Cube and sphere only need one number — the side length or the radius — so the height field is ignored. Cylinder, cone and square pyramid need two numbers: the radius (or base edge for the pyramid) and the perpendicular height. Type the dimensions in any unit you like, and the calculator returns the total surface area in that unit squared, along with the lateral (curved or sloped) surface and the base area separately.
How the calculation works
Each shape has a standard closed-surface formula. Cube: SA = 6a², the six identical square faces. Sphere: SA = 4πr², derived from the integral of the surface element over a sphere of radius r. Closed cylinder: SA = 2πr² + 2πrh — two circular ends plus the rectangular wrap that unrolls to width 2πr and height h. Closed cone: SA = πr² + πr·s, where s = √(r²+h²) is the slant height; πr·s is the lateral surface that unrolls to a circular sector. Square pyramid: SA = a² + 2a·s, where s = √((a/2)²+h²) is the slant height of each triangular face; the four triangles each have area ½·a·s, summing to 2a·s, plus the square base a². π is used at full machine precision throughout — only the displayed numbers are rounded.
Worked example
Take a cylinder with radius r = 3 and height h = 5. Base area is π·3² = 9π. Lateral area is 2π·3·5 = 30π. Total surface area is 30π + 2·9π = 48π ≈ 150.7964 square units. Check with a cone of the same radius and height h = 4. Slant height s = √(3²+4²) = √25 = 5. Lateral area πr·s = π·3·5 = 15π. Base area π·3² = 9π. Total 24π ≈ 75.3982 square units. Both fall out of the formulas directly with no rounding mid-calculation.
Frequently asked questions
What is the formula for the surface area of a sphere?
SA = 4πr², where r is the radius. A sphere has no flat face — every point on the surface is the same distance r from the centre — so the "lateral" and "total" surface area are the same number. For r = 5 the surface area is 4π·25 = 100π ≈ 314.159 square units.
What is the surface area of a cylinder?
For a closed right cylinder (both ends capped) the formula is SA = 2πr² + 2πrh. The 2πr² is the two circular ends and the 2πrh is the lateral surface that unrolls to a rectangle of width 2πr and height h. If the cylinder is open at one or both ends, drop the corresponding πr² terms.
How do I find the slant height of a cone or pyramid?
For a right circular cone, the slant height s is the hypotenuse of a right triangle with legs r (radius) and h (perpendicular height): s = √(r² + h²). For a right square pyramid, the slant height of the triangular face is s = √((a/2)² + h²), where a is the base edge. The calculator does both internally.
Does this include the base or not?
Yes — the formulas above give the closed surface area, meaning every face is included. The cube includes all six faces, the cylinder includes both circular ends, the cone includes the circular base, and the square pyramid includes the square base. For open-top variants (like an open tank), subtract the corresponding base term.
Why is the cone's lateral formula πr·s and not (1/2)·perimeter·slant?
It is — they are the same. The base perimeter of a cone is 2πr, half of that is πr, and multiplying by the slant height s gives πr·s. The (1/2)·perimeter·slant rule applies to any right pyramid or cone with a regular base; for a square pyramid with base edge a the perimeter is 4a, half is 2a, times s gives 2a·s.
What units does the answer come back in?
Whatever unit you entered, squared. Enter centimetres and the surface area is in cm². Enter inches and it is in in². The geometry is scale-invariant: doubling every dimension multiplies the surface area by four, which is why the formulas only depend on the ratios, not the absolute size.