Surface Area Calculator
Calculate surface area for 3D shapes including spheres, cylinders, cubes, cones, rectangular prisms, and pyramids.
Surface Area Result
Adding up every face
Surface area is the total area of the outside of a solid, in square units. For anything with flat faces it is exactly what it sounds like: find each face's area and add them. A rectangular box has three pairs of identical faces, which gives 2(lw + lh + wh) — a formula worth deriving once rather than memorising.
Curved solids follow from unrolling them. A cylinder's side is a rectangle whose width is the circumference, so the lateral area is 2πrh and the total adds two circular ends for2πr² + 2πrh. A cone's lateral area is πrl, where l is theslant height — not the vertical height, though the two relate through Pythagoras. A sphere is 4πr², exactly four times the area of its own great circle.
The relationship with volume is what makes surface area matter in practice. Volume grows with the cube of size while surface area grows with the square, so the surface-area-to-volume ratio falls as things get larger. That single fact explains why small animals lose heat faster, why crushed ice melts quicker than a block, and why cells divide rather than simply growing.
Frequently asked questions
What is surface area?
Surface area is the total area of the outer surface of a three-dimensional object.
What are the surface area formulas for common solids?
Sphere 4*pi*r^2; cylinder 2*pi*r(r+h); cone pi*r(r+l) where l is the slant height; cube 6a^2; rectangular box 2(lw+lh+wh). For a cone, the slant height is sqrt(r^2+h^2), which is the step most often missed.
What is the difference between total and lateral surface area?
Lateral surface area is the curved or side faces only; total surface area adds the bases. For a cylinder the lateral area is 2*pi*r*h while the total adds two circles of pi*r^2. Which you want depends on the job - labelling a tin needs lateral area, painting it inside and out needs total.
Why does surface area matter relative to volume?
Because surface area scales with the square of length while volume scales with the cube, so larger objects have proportionally less surface. That ratio governs heat loss, drag, dissolution rate and how quickly a cell can exchange nutrients - and it is why small animals lose heat faster and why crushed solids dissolve more quickly than lumps.