Surface Area Calculator
Compute the surface area of 13 common solids. Pick a shape, enter dimensions, and get total, lateral and base surface area separately with a live labeled diagram, unit conversions and full step-by-step working.
What is surface area?
Surface area is the total area of every outer face of a three-dimensional solid. If you could unfold the shape into flat pieces and lay them side by side, the combined area of those pieces is the surface area. It answers everyday questions like "how much paint do I need?", "how much wrapping paper covers this box?" and "how much sheet metal makes this tank?".
Because surface area is measured across two dimensions at once, it uses square units — square meters, square feet, square inches. Volume, by contrast, is cubic units. Doubling a shape's size multiplies its surface area by 4 and its volume by 8, which is why small animals lose heat quickly and large tanks are efficient to insulate.
Every formula on this page splits into two ideas: a lateral piece (the wrap-around sides) and a base piece (the flat top and/or bottom). Adding the two gives the total surface area. Reporting all three separately — the way this calculator does — is often more useful than one lump sum, because painters, fabricators and engineers usually care about just the sides or just the ends.
Surface area, shape by shape
One card per solid — the geometric idea, its formula with each variable spelled out, the diagram you see above the results panel, and one hand-checked numeric example.
Sphere
A perfectly round ball. Its surface area is exactly four times the area of the great circle that cuts through its centre.
Cube
Six identical square faces. Because every face has the same area s², the total surface is simply six of them added together.
Cylinder
Two circular ends and one rectangular side. The side unrolls into a rectangle of width 2πr (the circumference) and height h.
Cone
One circular base and a curved side that unrolls into a circular sector. The slant height ℓ = √(r² + h²) is the sector's radius.
Rectangular tank (box)
Three pairs of matching rectangles. Every pair contributes twice, giving the classic 2(lw + lh + wh) sum.
Capsule
A cylinder capped with a hemisphere on each end. The two hemispheres combine into a full sphere.
Spherical Cap
The dome you get by slicing a sphere with a flat plane. The curved dome uses 2πRh; add the flat circular base πa² if you want the whole closed cap, where a² = h(2R − h).
Conical Frustum
A cone with its tip sliced off flat. The two circular ends contribute π(r₁² + r₂²); the slanted side unrolls into a ring-shaped strip of area π(r₁ + r₂)ℓ.
Ellipsoid
A stretched sphere with three independent semi-axes. There's no exact closed-form formula, so this calculator uses Knud Thomsen's approximation (p = 1.6075), accurate to within about 1.06%.
Pyramid (rectangular base)
One rectangular base and four triangular sides meeting at a single apex. Opposite triangles are congruent, so the four lateral faces reduce to two pairs.
Triangular Prism
Two identical triangular ends joined by three rectangles. The two triangles contribute 2A and the three side rectangles together contribute P × L (triangle perimeter times prism length).
Tube (Hollow Cylinder)
A pipe: an outer cylinder wall, an inner cylinder wall, and two ring-shaped ends. Using outer radius R = D/2 and inner radius r = d/2, the outside contributes 2πRL, the inside 2πrL, and the two annular ends 2π(R² − r²).
Torus
A donut swept out by a small circle of radius r moving around a bigger circle of radius R. Its surface is (circle circumference) × (sweep circumference).
Common area unit conversions
| From | To | Multiply by |
|---|---|---|
| 1 square meter (m²) | square feet | 10.7639 |
| 1 square meter (m²) | square yards | 1.19599 |
| 1 square meter (m²) | square inches | 1550.003 |
| 1 square meter (m²) | square centimeters | 10000 |
| 1 square foot (ft²) | square meters | 0.092903 |
| 1 square foot (ft²) | square inches | 144 |
| 1 square yard (yd²) | square meters | 0.836127 |
| 1 square inch (in²) | square centimeters | 6.4516 |
| 1 square centimeter (cm²) | square inches | 0.15500 |
Features of this calculator
- 13 solids — sphere, cube, cylinder, cone, rectangular tank, capsule, spherical cap, conical frustum, ellipsoid, pyramid (rectangular base), triangular prism, tube (hollow cylinder) and torus.
- Reports total, lateral and base surface area separately for every shape where the distinction is meaningful — so painters, fabricators and engineers get the exact number they need.
- Live SVG diagram for every shape with each entered dimension labeled directly on the figure.
- Spherical cap accepts any two of {ball radius R, cap height h, base radius a} and derives the third from a² = h(2R − h).
- Triangular prism supports right-triangle (base + height) or any-triangle (three sides + Heron's formula) input, with a full perimeter derivation.
- Ellipsoid uses Knud Thomsen's p = 1.6075 approximation — accurate to about 1.06% across all triaxial ellipsoids.
- Length unit selector (mm, cm, m, km, in, ft, yd); every result is instantly reshown in m², ft², yd², in² and cm².
- User-selectable significant figures (3 to 10) so the result matches the precision you need.
- Full step-by-step working — formula, slant-height / perimeter derivations, substitution and total shown for every shape.
- Copy the result or download it as a PNG straight from the results panel.
- Cross-links to the Volume Calculator and Area Calculator for shape-by-shape comparisons.
Frequently asked questions
+What is surface area?
Surface area is the total area of all the outer faces of a three-dimensional object — the amount of material you would need to wrap it exactly. It's measured in square units (m², ft², in², cm², yd²).
+What's the difference between total, lateral and base surface area?
Lateral surface area covers the side faces only. Base surface area covers the flat top and/or bottom. Total surface area is the sum of both. This calculator reports all three separately for every shape where the distinction is meaningful.
+What is the surface area of a sphere?
S = 4πr², where r is the radius. A sphere has no distinct base, so total and lateral surface area are the same.
+How do I find the surface area of a cylinder?
Add the two circular ends and the rectangle you would get by unrolling the side: S = 2πr² + 2πrh = 2πr(r + h).
+How do I calculate the surface area of a cone?
S = πr² + πrℓ, where ℓ = √(r² + h²) is the slant height. The lateral piece πrℓ is the cone's curved side unrolled into a sector.
+What's the lateral surface area of a rectangular tank?
The four side walls only: 2h(l + w). The top and bottom (l × w each) are the two bases. Total = 2(lw + lh + wh).
+How do I find the surface area of a triangular prism?
S = 2A + PL, where A is the triangle's area, P is the triangle's perimeter, and L is the prism's length. This calculator accepts either right-triangle (base + height) or any-triangle (three sides via Heron's formula) input.
+How is the surface area of an ellipsoid calculated?
An ellipsoid has no closed-form surface area. This calculator uses Knud Thomsen's approximation with p = 1.6075, which is accurate to within about 1.061% for any triaxial ellipsoid.
+What's the formula for the surface area of a torus?
S = 4π²Rr, where R is the distance from the torus centre to the tube centre and r is the tube radius.
+Can I switch units and precision?
Yes — pick any length unit (mm, cm, m, km, in, ft, yd) and any significant-figures level (3 to 10). Every result is instantly reshown in m², ft², yd², in² and cm².