Surface Area Calculator
Surface area and volume of a cube, box, sphere, cylinder, cone, square pyramid, capsule or torus, in any unit, with the formula and the arithmetic shown.
Surface Area Calculator: with the default inputs, total surface area is 7.853982 m².
Only the dimensions this shape needs are read; the rest are ignored.
Cube edge, box length, or the base edge of a square pyramid.
Box only.
Box, cylinder, cone and pyramid. For a capsule it is the straight middle section, not counting the two domed ends.
Sphere, cylinder, cone and capsule. For a torus this is the tube (minor) radius.
Torus only: from the centre of the ring to the centre of the tube. Must exceed the tube radius.
- Volume
- 1.570796 m³
- Lateral (curved / side) area
- 6.283185 m²The sides only, with the flat ends left off. Not defined for a closed curved surface.
- Base area
- 0.785398 m²
- Surface area ÷ volume
- 5Per meter. High means it loses heat, dries out or reacts fast for its size.
- Formula used
- SA = 2πr² + 2πrh = 2πr(r + h), V = πr²h
Assumptions
- Closed solids with no wall thickness — the area is the outer surface only.
- The pyramid is right and has a square base; the cone and cylinder are right circular.
- A torus is a ring torus, so the major radius must exceed the tube radius.
How this is worked out
The formula
Cube SA = 6a² V = a³ Box SA = 2(ab + ah + bh) V = abh Sphere SA = 4πr² V = 4πr³ ÷ 3 Cylinder SA = 2πr(r + h) V = πr²h Cone SA = πr(r + l), l = √(r² + h²) V = πr²h ÷ 3 Square pyramid SA = a² + 2al, l = √((a/2)² + h²) V = a²h ÷ 3 Capsule SA = 2πr(2r + h) V = πr²h + 4πr³ ÷ 3 Torus SA = 4π²Rr V = 2π²Rr²
Open How it’s calculated above to see this worked through with your own numbers.
What you enter
- Shape
- Only the dimensions this shape needs are read; the rest are ignored.Cube · Box (rectangular prism) · Sphere · Cylinder · Cone · Square pyramid · Capsule (cylinder with hemisphere ends) · Torus (ring / doughnut)
- Edge / length a
- Cube edge, box length, or the base edge of a square pyramid.in mm, cm, m, km, in, ft, yd · 0 or more · defaults to 1
- Width b
- Box only.in mm, cm, m, km, in, ft, yd · 0 or more · defaults to 1.5
- Height h
- Box, cylinder, cone and pyramid. For a capsule it is the straight middle section, not counting the two domed ends.in mm, cm, m, km, in, ft, yd · 0 or more · defaults to 2
- Radius r
- Sphere, cylinder, cone and capsule. For a torus this is the tube (minor) radius.in mm, cm, m, km, in, ft, yd · 0 or more · defaults to 0.5
- Major radius R
- Torus only: from the centre of the ring to the centre of the tube. Must exceed the tube radius.in mm, cm, m, km, in, ft, yd · 0 or more · defaults to 1
What you get back
- Total surface areamain answer
- Volume
- Lateral (curved / side) area
- The sides only, with the flat ends left off. Not defined for a closed curved surface.
- Base area
- Surface area ÷ volume
- Per meter. High means it loses heat, dries out or reacts fast for its size.
- Formula used
What this assumes
- Closed solids with no wall thickness — the area is the outer surface only.
- The pyramid is right and has a square base; the cone and cylinder are right circular.
- A torus is a ring torus, so the major radius must exceed the tube radius.
About this calculator
Surface area is what you paint, plate, wrap, insulate or lose heat through; volume is what you fill. They scale differently — double every dimension and the area quadruples while the volume goes up eightfold — which is why this calculator reports both, plus the ratio between them.
How to use it
Pick a shape and fill in only the dimensions it names; everything else is ignored. Every length field has its own unit dropdown, so you can enter a radius in inches and read the area in square feet. Results come back in whatever units you choose for the outputs.
The lateral area is the side surface with the flat ends left off — the label on a tin, the wall of a silo, the sloping face of a cone. It is what you want for wrapping or cladding, where the ends are open or already covered. Sphere and torus have no ends, so the field is blank for them.
The shapes that are worth a note
A cone's curved surface is πrl, using the slant height l = √(r² + h²), not the vertical height. Using h by mistake is the single most common error in cone problems, and it always undershoots.
A square pyramid has the same trap: its slant height runs up the middle of a triangular face, from the midpoint of a base edge to the apex, so it is √((a/2)² + h²) — not the edge from a corner to the apex, which is longer.
A capsule is a cylinder capped with a hemisphere at each end, the shape of a pressure vessel or a pill. Enter the straight section as the height; the two domes together make exactly one sphere, so the total length end to end is h + 2r.
A torus is a doughnut. R is the major radius, from the centre of the hole to the centre of the tube; r is the tube itself. Both results fall out of Pappus's centroid theorems: the surface area is the tube's circumference times the distance its centre travels round the ring, and the volume is the tube's cross-section area times the same distance. R must exceed r, or the shape self-intersects and the formulas stop being true.
Surface-area-to-volume ratio
The last output divides one by the other. It has units of 1/length, and it explains a surprising amount of the physical world: why crushed ice melts faster than a block, why small animals eat more per gram than large ones, why a sphere is the shape that minimises heat loss for a given volume, and why cells stay microscopic. Halve the size of anything and you double this ratio.
Precision
Everything is computed in double precision from exact closed forms, so the only error is the accuracy of the dimensions you type. Very small or very large values in tiny units can overflow when cubed; if that happens the calculator says so rather than returning a meaningless number.
Frequently asked questions
▸What is the surface area of a cylinder?
2πr(r + h): the side unrolls into a rectangle of width 2πr and height h, and the two circular ends add πr² each. For a radius of 0.5 m and a height of 2 m that is 7.854 m².
▸Why does a cone use the slant height?
Because the curved surface, flattened out, is a sector of a circle whose radius is the slant l = √(r² + h²), not the vertical height. Using h instead always gives an answer that is too small.
▸What is lateral surface area?
The sides only, with the flat top and bottom left off — the label area of a tin, or the cladding on a silo. Total surface area adds the ends back on.
▸Why does surface-area-to-volume ratio matter?
Anything that happens at a surface — heat loss, evaporation, absorption, chemical reaction — scales with area, while the mass involved scales with volume. Small things have far more surface per unit of volume, which is why they cool, dry and react faster.
▸How is this different from the volume calculator?
The volume calculator leads with volume for the six common solids. This one leads with surface area and adds the capsule and the torus, plus lateral area, base area and the area-to-volume ratio.
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