Surface Area Calculator
Pick a shape, enter the dimensions in your preferred units (mm, cm, m, in, ft, yd, etc.), and instantly see the surface area in m², ft², in², cm² or any other area unit. Includes the formula and a working breakdown for every shape.
Quickly work out the surface area of any common 3D shape — sphere, cube, cone, cylinder, rectangular tank, capsule, spherical cap, conical frustum, ellipsoid or square pyramid.
Mix and match length units per input (millimetres, centimetres, metres, kilometres, inches, feet, yards or miles) and pick the area unit you want for the result (mm², cm², m², km², in², ft², yd², mi², acres or hectares).
Each shape shows the formula being used and a clear breakdown — the curved surface, the base disc, the lateral area, the end caps — so you can sanity-check the maths and not just trust the number.
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What this calculator does
Computes the total surface area of 10 common 3D shapes from your input dimensions, with full unit support on every input and on the result. Shows the formula used and a labelled breakdown (e.g. base + lateral) wherever it makes sense.
Who it is for
Students, teachers, DIYers, builders, fabricators, packaging designers, architects, painters and decorators estimating coverage, anyone designing tanks, silos, capsules, prototypes or 3D-printed parts, and anyone who needs a quick, accurate surface-area answer without doing the algebra by hand.
How to use it
Pick a shape from the grid. Enter the dimensions and choose a unit for each (you can mix units — e.g. radius in inches, height in feet). Choose the area unit for your answer. The result updates instantly with the surface area, the formula used and a breakdown of how it splits across the shape's faces.
How the calculation works
Sphere SA = 4πr². Cube SA = 6a². Cone SA = πr² + πr√(r²+h²). Closed cylinder SA = 2πr² + 2πrh. Rectangular tank SA = 2(lw + lh + wh). Capsule SA = 4πr² + 2πrh. Spherical cap SA = 2πRh + πr² (provide any two of base radius r, ball radius R, cap height h). Conical frustum SA = π(r₁+r₂)·s + πr₁² + πr₂² where s = √((r₂−r₁)² + h²). Ellipsoid SA ≈ 4π·((aᵖbᵖ + aᵖcᵖ + bᵖcᵖ)/3)^(1/p) with p = 1.6075 (Knud Thomsen approximation). Square pyramid SA = a² + 2a·√((a/2)² + h²).
Worked example
A closed cylindrical tank with base radius 1.2 m and height 3 m: ends = 2π × 1.2² ≈ 9.048 m². Side = 2π × 1.2 × 3 ≈ 22.619 m². Total surface area ≈ 31.667 m² (≈ 340.86 ft²).
Assumptions and limitations
- All shapes assume a smooth mathematical surface and a closed solid (cylinder, rectangular tank, capsule, square pyramid all include all faces).
- Spherical-cap calculations require any two of: base radius r, ball radius R, cap height h. The third value is derived from r² = h(2R − h).
- Ellipsoid surface area uses the Knud Thomsen approximation (p = 1.6075). Maximum error is roughly 1.06% — accurate enough for almost all practical uses.
- Length unit conversions use exact factors (1 in = 0.0254 m, 1 ft = 0.3048 m, 1 yd = 0.9144 m, 1 mi = 1609.344 m).
- Area outputs convert from m² using the same exact factors squared.
Frequently asked questions
What is the surface area of a sphere?
The surface area of a sphere is 4πr², where r is the radius. For a sphere with a 1 m radius, SA = 4π × 1² ≈ 12.566 m².
How do I calculate the surface area of a cylinder?
For a closed cylinder, SA = 2πr² + 2πrh — the two circular ends plus the rectangular side wrapped around. The calculator shows both parts separately so you can see the split.
What is the formula for the surface area of a cube?
A cube has 6 identical square faces, so SA = 6a², where a is the edge length. For a 4-inch cube, SA = 6 × 16 = 96 in².
How do I work out the surface area of a rectangular tank?
A closed rectangular tank (cuboid) has SA = 2(lw + lh + wh) — the sum of all six rectangular faces.
Can I mix units?
Yes. You can enter the radius in inches and the height in feet, for example, and pick the result in m² or ft². The calculator converts everything to metres internally before applying the formulas.
How accurate is the ellipsoid surface area?
Ellipsoid surface area has no closed-form solution, so the calculator uses the Knud Thomsen approximation with p = 1.6075. Maximum error is around 1.06% — well below what matters in any practical setting.
What is a spherical cap?
A spherical cap is the rounded 'dome' you get when you slice a sphere with a flat plane. It is described by any two of: base radius r, ball radius R, and cap height h. The calculator works out the third value automatically.
What is a conical frustum?
A conical frustum is what you get when you cut the top off a cone with a plane parallel to the base — like a bucket or a paper cup shape. Its surface area combines the slanted sides, the small top circle and the larger base circle.
Why do you call it a capsule?
A capsule is a cylinder with hemispherical ends, like a tablet. Its surface area equals the surface of a full sphere (4πr²) plus the cylindrical side wall (2πrh).
Is the surface area of an open cylinder different?
Yes. An open cylinder (no lids) is just the side wall: 2πrh. A cylinder with one lid would be πr² + 2πrh. This calculator assumes a closed cylinder; subtract the relevant πr² values if your shape is open.
Related calculators
- Annulus Area Calculator — Enter the outer and inner radii. The calculator returns A = π·(R² − r²).
- Area Calculator — Pick a 2D shape — circle, square, rectangle, triangle, trapezium, parallelogram, ellipse, regular polygon, rhombus, sector, annulus or kite
- Capsule Surface Area Calculator — Enter the radius and the straight middle length. The calculator returns 4πr² for the two rounded ends plus 2πrh for the cylindrical side.
- Circle Area Calculator — Enter the radius. The calculator returns A = πr², the area of the flat disc.
- Circular Sector Area Calculator — Enter the radius and the angle in degrees. The calculator returns the sector's area: A = ½·r²·θ (with θ converted to radians).
- Cone Surface Area Calculator — Enter the base radius and height. The calculator splits the answer into the base (πr²) and the lateral surface (πr·s), where s = √(r²+h²) is
Related guides
- Surface Area Formulas for Common 3D Shapes — A clear, plain-English guide to surface area formulas for sphere, cube, cone, cylinder, rectangular tank, capsule, spherical cap, conical frustum, ellipsoid and
Sources and references
Standard geometric surface area formulas (engineering and mathematics textbooks). Knud Thomsen ellipsoid approximation. SI and Imperial unit conversions per the International System of Units (SI) and NIST.
Last updated
Last reviewed: 2026-04-21.