Surface Area Calculator

Surface area is the total area of every outer face of a three-dimensional object — the amount of skin it would take to cover it completely. It is always measured in square units such as cm², m² or in², in contrast to volume, which fills the inside and is measured in cubic units. This calculator finds the surface area of the five shapes you meet most often — sphere, cylinder, cone, cube and rectangular prism (box) — from a few simple measurements. It is useful for school and college geometry, but also for very practical jobs: estimating how much paint covers a tank, how much sheet metal wraps a duct, how much wrapping paper or laminate a box needs, or how much heat a body can shed.

Formulas by shape

Each shape has its own closed-form formula derived from its geometry. In every formula below, π (pi) is about 3.14159, r is a radius, h is a height, s is a cube's side, and l, w, h are the length, width and height of a box.

• Sphere: SA = 4πr² — a perfectly round ball has no flat faces, just one curved surface set by its radius.
• Cylinder (with both caps): SA = 2πr(r + h) — the term 2πr² is the two circular ends and 2πrh is the curved side wrapped around them.
• Cone (with base): SA = πr(r + slant), where the slant height = √(r² + h²) from the Pythagorean theorem; πr² is the circular base and πr·slant is the curved cone surface.
• Cube: SA = 6s² — six identical square faces, each of area s².
• Rectangular prism: SA = 2(lw + lh + wh) — three pairs of matching rectangular faces.

Worked example

Take a sphere of radius 5 cm. Square the radius to get 25, multiply by π to get about 78.54, then by 4: the surface area is 4π(25) ≈ 314.16 cm². For a cube with side 4 cm, each face is 4 × 4 = 16 cm² and there are six of them, so SA = 6 × 16 = 96 cm². For a closed cylinder with radius 3 cm and height 10 cm, SA = 2π × 3 × (3 + 10) = 78π ≈ 245.04 cm². And a cone with radius 3 cm and height 4 cm has slant √(9 + 16) = 5, so SA = π × 3 × (3 + 5) = 24π ≈ 75.40 cm². Enter the same numbers above to see each result and its formula breakdown.

Total versus lateral surface area

For shapes with flat ends — the cylinder and the cone — it helps to know whether you need the total surface area or only the lateral (side) surface area. The total includes the end caps; the lateral covers just the curved wall. A closed cylinder is 2πr(r + h), but a label wrapped only around its body uses the lateral area 2πrh, leaving out the two circles. Likewise a cone's total is πr(r + slant), while the cloth forming just its slanted side is πr·slant without the base. The formulas above and the calculator return the full closed surface, so subtract the cap you do not need when an object is open at one end.

Where surface area is used

Surface area drives a surprising number of real-world decisions. Manufacturers use it to price the metal, plastic or fabric needed to make a part; painters and builders use it to estimate litres of paint or square metres of cladding for a tank, wall or roof. In chemistry a finely divided solid reacts faster because it exposes far more surface area for the same volume, which is why powders dissolve quicker than lumps. Engineers sizing a radiator or heat sink maximise surface area to shed heat, and in medicine drug doses are often scaled to a patient's body surface area rather than weight alone. Even packaging designers minimise surface area to cut material cost while still enclosing the required volume.

Tips

• Decide first whether you need the full closed surface or only part of it — an open-top cylinder or tank drops the missing cap.
• For a cone, the formula needs the slant height; the calculator derives it from the vertical height you enter, so supply the upright height, not the slope.
• Sketch the shape and label each measurement before typing it in, so radius and height are never swapped.

Common mistakes

• Mixing units — convert every dimension to the same unit first, or the square-unit answer will be meaningless.
• Confusing radius with diameter; the radius is half the diameter, and using the diameter doubles every radius term.
• Using the cone's vertical height in place of the slant height — they are equal only when the cone is flat, which it never is.
• Reporting surface area in cubic units; surface area is always squared, volume is cubed.