Surface Area Calculator
Calculate the surface area of a box, cylinder, sphere, cone, or capsule, split into lateral and end areas, with every common unit conversion.
Comment utiliser cette calculatrice
- 1Choose the solid you are measuring.
- 2Pick a unit and use it for every dimension.
- 3Enter the dimensions — for a cone, enter the vertical height and the calculator derives the slant height for you.
- 4Use the total for full coverage, or the lateral figure when the ends are not being covered.
Comment ça marche
Surface area formulas
Box A = 2(lw + lh + wh) Cylinder A = 2πrh + 2πr² Sphere A = 4πr² Cone A = πrl + πr², l = √(r² + h²) Capsule A = 2πrh + 4πr²
Every formula splits into a lateral part and an end part. Unroll a cylinder's side and it becomes a rectangle of width 2πr and height h; the two circular ends add 2πr². A cone's side unrolls into a sector whose area is πrl, where l is the slant height found by Pythagoras from the radius and vertical height — not the vertical height itself.
Exemple détaillé
A cylindrical tank with a 3 ft radius and an 8 ft height has a side area of 2π × 3 × 8 = 150.80 ft² and two ends of 2π × 3² = 56.55 ft², giving a total surface area of 207.35 square feet.
Surface Area Calculator : le guide complet
Why surface area matters more than volume for coverage
Paint, insulation, plating, wrapping, and corrosion all act on the outside of an object, so surface area — not volume — determines how much material you need. A one-gallon can of paint typically covers 350 to 400 square feet on a smooth, primed surface, so dividing the total surface area by 350 gives a realistic first estimate.
Rough or porous surfaces absorb considerably more. Bare masonry, textured render, and unprimed wood can cut coverage in half, and a second coat is usually needed regardless — so doubling the estimate is the safe default for anything other than a repaint over a sealed surface.
Slant height versus vertical height on a cone
A cone's curved surface is described by its slant height, the distance from the tip down the outside to the edge of the base. That is longer than the vertical height, and using the vertical height instead always understates the area.
The two are linked by Pythagoras: l = √(r² + h²). For a cone with a 4 ft radius and a 9 ft height, the slant height is √(16 + 81) = 9.85 ft — nearly a foot longer than the vertical measurement. This calculator derives it automatically, and reports it so you can check against a tape measure.
The square-cube law
Scale a solid up by a factor of two and its surface area grows fourfold while its volume grows eightfold. This is the square-cube law, and it explains a surprising amount of the physical world.
It is why large animals overheat more easily than small ones, why crushed ice melts faster than a block of the same mass, and why chemical reactors are designed with internal coils rather than relying on the vessel wall. Practically, it means a large tank costs proportionally less to insulate per unit of contents than a small one.
Questions fréquentes
What is the difference between total and lateral surface area?
Lateral surface area is just the side — the curved wall of a cylinder or the sloping face of a cone. Total surface area adds the flat ends or base. Use the lateral figure when the ends are not being treated, for example a tank sitting on a concrete pad or a label wrapped around a can.
How much paint do I need for this area?
Divide the surface area in square feet by the coverage on the paint can, typically 350–400 ft² per gallon for one coat on a smooth surface. Add a second coat, and expect noticeably lower coverage on rough or unprimed surfaces.
How do I find the surface area of a cube?
Choose the box option and enter the same value for length, width, and height. All six faces are identical, so the total is 6s² — a 3-inch cube has a surface area of 54 square inches.
Why is a sphere's surface area exactly four times its cross-section?
A sphere of radius r has surface area 4πr², while the flat circle through its centre has area πr². The exact factor of four was proved by Archimedes, who considered it his finest result and asked for the diagram to be carved on his tomb.