This surface area calculator finds the total surface area of the five core solids — sphere, cylinder, cone, cube and rectangular prism — from their dimensions, in one place.
Five Solids, Five Ways of Unfolding
Every surface-area formula is a net unfolded: cylinders unroll to rectangles + circles, cones to sectors. Capacity partners with skin on our Volume Calculator. Part of our Math Calculators collection.
Worked Examples (Verified)
- Sphere r = 5 (defaults): 4π × 25 = 314.16 — exactly 4× the great circle’s area, Archimedes’ favorite theorem
- Cylinder r = 3, h = 7: 2π × 3 × 10 = 188.50 (lateral 131.95 + two caps 56.55)
- Cone r = 3, h = 4 (slant 5): π × 3 × 8 = 75.40 — the 3-4-5 triangle doing quiet work again
- Cube a = 4: 6 × 16 = 96; prism 2×3×4: 2(6+8+12) = 52
Why Engineers Live Here
- Paint/coating estimates start as surface area × coats ÷ coverage — with waste factors for texture.
- Heat loss tracks skin: insulation R-value math needs wall/roof areas (house-shaped prisms, mostly).
- Biology’s limits are square-cube: cells exchange across surface but consume by volume — size ceilings everywhere in nature.
- Packaging material cost is surface area with a profit margin attached.
Related Tools
- Volume Calculator — the inside to this outside
- Cone Calculator — cone in full detail
- Math Calculators — the full collection
Frequently Asked Questions
How do you find the surface area of a sphere?
4πr² — for r = 5, 314.16 square units. Remarkably, that equals the lateral area of the sphere’s circumscribing cylinder (Archimedes proved it and wanted it on his tombstone). No caps or deductions: a sphere’s surface is all lateral.
What is the surface area of a cylinder?
2πr(r + h) for a closed cylinder: lateral area 2πrh plus two circular caps 2πr². For r = 3, h = 7: 188.50. Open-top tanks subtract one cap (πr²); pipes use lateral only — this calculator assumes the standard closed solid, so adjust for open variants.
How is a cone’s surface area calculated?
πr(r + ℓ) where ℓ = √(r² + h²) is the slant height: base πr² plus lateral πrℓ (the unrolled sector). r = 3, h = 4 gives slant 5 and area 75.40. Using h instead of ℓ in the lateral term is the classic error — the formula wants the slope, not the altitude.
Surface area vs volume — what’s the difference?
Area (units²) measures skin; volume (units³) measures capacity. Doubling a cube’s edge quadruples surface (×4) but octuples volume (×8) — the square-cube law. Painters buy area; tankers buy volume; this page and the Volume Calculator are a matched pair.
Can I use this for a room’s paint area?
A rectangular room is a prism: walls = perimeter × height (2(l+w)h) — compute the prism, then subtract ceiling/floor and openings as needed. Dedicated coverage math (gallons, coats) lives on the Paint Calculator in the Construction hub.
Is this surface area calculator free?
Yes — free, instant, no sign-up.