Surface Area Calculator
Pick a solid and the fields adapt: sphere, cone, cube, cylinder, rectangular tank, capsule, or square pyramid. You get the total surface area plus the lateral and base breakdown, with each formula and substitution written out.
Substitution
Total = lateral + bases
Surface area is the sum of every face. For solids with flat tops or bottoms — cones, cylinders, pyramids, tanks — it splits naturally into the wrap-around lateral area plus the base(s), and this calculator reports each part so you can drop the base when it doesn’t apply (painting a silo that sits on concrete, say). Curved solids lean on π: a sphere is exactly four great circles (4πr²), a cone’s slant unrolls into a fan of πrℓ, and a capsule is a cylinder wrap plus a full sphere from its two end caps.
How it’s calculated
Sphere: 4πr². Cone: πrℓ + πr² with slant ℓ = √(r²+h²). Cube: 6a². Cylinder: 2πrh + 2πr². Rectangular tank: 2(lw + lh + wh). Capsule: 2πrh + 4πr² (h = cylinder section only). Square pyramid: a² + 2a√(h² + a²/4). Units are whatever you enter, squared.
Results update as you type and are for education, not professional advice — double-check any number that matters.
Worked example
A sphere with r = 3 has 4π × 9 = 113.10. A cone with r = 3, h = 4: slant ℓ = 5, so π×3×5 + π×9 = 47.12 + 28.27 = 75.40. A 2 × 3 × 4 tank: 2(6 + 8 + 12) = 52. A capsule with r = 2, h = 5: 2π×2×5 + 4π×4 = 62.83 + 50.27 = 113.10. A pyramid with base 4, height 6: 16 + 8√40 = 66.60.
Common mistakes
- Using the vertical height in the cone formula — the lateral term needs the slant ℓ = √(r²+h²), which the tool derives for you.
- Forgetting that the capsule height here is the cylinder section only — total end-to-end length is h + 2r.
- Doubling the base for solids that have one — a cone and a pyramid have a single base; a cylinder has two.
Where it is used
- Paint, coating, and insulation estimates for tanks, silos, and ducts.
- Wrapping, laminating, or plating costs proportional to surface.
- Geometry homework on composite solids.
Frequently asked questions
Which measurements does each solid need?
Sphere: radius. Cube: one edge. Cone, cylinder, capsule: radius + height. Rectangular tank: length, width, height. Square pyramid: base edge + vertical height. The fields relabel themselves when you switch solids.
What is the difference between lateral and total surface area?
Lateral covers only the sides — the wrap. Total adds the base(s). Painting the outside of a standing cylinder that sits on the ground needs the lateral area plus one base, not two; the breakdown rows let you assemble exactly what you need.
How is the pyramid’s slant height found from its vertical height?
By a right triangle from the apex to the midpoint of a base edge: slant = √(h² + (a/2)²). For base 4 and height 6 that is √(36 + 4) = 6.3246, giving lateral area 2a × slant = 50.60.
Why does a capsule use 4πr² even though the caps are separated?
The two hemispherical ends always combine into one complete sphere of the same radius — so a capsule is exactly one sphere plus one open cylinder, whatever its length.
A sphere and a capsule both show 113.10 here — coincidence?
A neat one from these particular numbers: 4π·3² = 36π, and 2π·2·5 + 4π·2² = 20π + 16π = 36π. Different shapes, same total.