Surface area of every common 3D shape — cube, rectangular box, sphere, cylinder, cone, square pyramid, hemisphere — with formula, working, and a labelled diagram.
Pick a shape, type its dimensions, read the area.
Total surface area
Lateral area
54
sides only
Base area
20
one base
Volume
60
cubic units
SA : V ratio
1.57
per unit volume
| Shape | Total surface area | Volume |
|---|---|---|
| Cube (side a) | 6 a² | a³ |
| Rectangular box (a, b, c) | 2(ab + bc + ca) | abc |
| Sphere (radius r) | 4π r² | ⁴⁄₃ π r³ |
| Cylinder (r, h) | 2πr² + 2πrh | π r² h |
| Cone (r, h) | π r² + π r ℓ | ⅓ π r² h |
| Hemisphere (r) | 3 π r² | ⅔ π r³ |
| Square pyramid (a, h) | a² + 2 a ℓ | ⅓ a² h |
The Method
Surface area is the total area of the outer skin of a 3D object. For shapes with flat faces (cube, box, pyramid) you add the area of every face. For shapes with curved surfaces you use a closed-form formula derived from calculus — for example a sphere has 4πr², exactly four times the area of its "great circle". Total surface area includes the bases; lateral surface area excludes them. The cylinder is the cleanest example: unroll the curved side into a rectangle of width 2πr and height h, and add the two circular ends.
Working for your shape
About This Tool
A surface area calculator finds the total external area of a 3D solid — the amount of "skin" wrapping it. This tool covers the seven shapes you meet most often: cube, rectangular box (cuboid), sphere, cylinder, cone, hemisphere and square pyramid. For each it returns the total surface area, the lateral surface area, the base area, the volume, and the surface-to-volume ratio.
Surface area is the answer to practical questions about cost of paint or coating, amount of wrapping paper, heat loss through a container's walls, the strength of a beam against bending, and how fast a chemical reaction proceeds (rate depends on surface area). It also matters in biology — cells stay small because their surface-to-volume ratio determines how fast they can exchange nutrients and waste with their surroundings.
Mathematically, surface area is the result of integrating over a 2D surface in 3D space. For the simple shapes here, closed-form formulas exist — derived once and reused forever. The total surface area of a sphere, 4πr², was famously proved by Archimedes around 250 BCE, who also showed that a sphere's surface area equals the lateral area of its circumscribed cylinder.
Use this free surface area calculator for geometry homework, packaging design, DIY projects (how much paint do I need?), or science class. All calculation runs locally in your browser — no sign-up, no tracking.
7 Common Shapes
Cube, box, sphere, cylinder, cone, hemisphere, square pyramid — pick from a dropdown.
Lateral & Total
Both kinds of surface area shown side-by-side — for open vs. closed containers.
Volume Too
Volume is calculated alongside surface area — useful for paint coverage and capacity.
SA : V Ratio
Surface-to-volume ratio — critical in biology, chemistry and heat transfer.
100% Free & Private
No account, no tracking — every calculation runs locally in your browser.
Step-by-Step Working
The formula and arithmetic for the chosen shape — perfect for homework.
Pick a shape, type its dimensions, read the total — with a diagram.
Choose from the dropdown — cube, box, sphere, cylinder, cone, hemisphere, or square pyramid. The input fields and formula update for the chosen shape.
Field labels change with the shape — side a for a cube, radius r for a sphere, r & h for a cylinder. Use any consistent units.
The headline number is the total surface area in square units of whatever you typed. The meta line shows the formula and substitution.
For open-top containers, you only need the lateral area plus one base — both are reported automatically.
A labelled diagram of the chosen shape appears in the chart panel — useful to verify you've assigned each dimension correctly.
Open the formula card to see the full substitution — perfect for showing your work in a homework or exam answer.
Everything you need to know about surface area, total vs lateral, and the common formulas.
Surface area is the total area of every outer face of a 3D object — the area of "wrapping" you'd need to cover the solid exactly. It is measured in square units of whatever length unit you use (cm², m², in²). Don't confuse it with volume, which is the amount of space inside the solid (cubic units).
Total surface area (TSA) includes every face. Lateral surface area (LSA) excludes the bases — it's just the side(s). For a cylinder, TSA = 2πr² + 2πrh while LSA = 2πrh. Use LSA when you only need the curved side (e.g. label area on a tin) and TSA when you need everything (e.g. paint for a closed box).
Surface area is always in square units matching your input. If you typed dimensions in cm, the result is cm²; in m, it's m²; in inches, it's in². The calculator doesn't assume a unit — it returns a pure number which you label appropriately.
SA = 4πr² — exactly four times the area of its great circle (the equator). This famous result was proved by Archimedes around 250 BCE, who also showed that a sphere's surface area equals the lateral area of its circumscribed cylinder (a result so beloved it was carved on his tomb).
TSA = 2πr² + 2πrh. The first term is the two circular ends; the second is the curved side, which "unrolls" into a flat rectangle of width 2πr (the circumference) and height h. Visualising the unroll is the easiest way to remember the formula.
Slant height ℓ = √(r² + h²) — the distance from a point on the base circle to the apex, measured along the curved surface. It comes straight from the Pythagorean theorem applied to the right triangle formed by the radius, the height, and the slant. The lateral area of a cone is πrℓ, not πrh.
The surface-to-volume (SA:V) ratio is surface area divided by volume. Smaller objects have a larger SA:V ratio — which is why cells stay tiny (they need surface for nutrient exchange) and why crushed ice melts faster than a single ice cube. For a sphere of radius r, SA:V = 3 / r, so doubling the radius halves the ratio.
Take the total surface area, subtract any faces you won't paint (e.g. the bottom of a box), and divide by the paint's coverage rating (e.g. 10 m² per litre). Add a small allowance (10–15%) for waste and a second coat. For curved surfaces, use the formula's curved-side term, not just a 2D projection.
Yes — any positive real number works. Decimals are fully supported (e.g. 2.5, 0.75). For fractions, convert to decimal first. Results use IEEE 754 double-precision arithmetic, accurate to about 15–17 significant digits.
This calculator covers the seven standard solids you meet in geometry class and most engineering problems. For irregular shapes, decompose into the basic solids and add the areas (subtracting any internal faces that no longer exist). For genuinely freeform shapes, use a CAD package — they can integrate surface area over arbitrary geometry.
Of all shapes with a given volume, the sphere has the smallest surface area — the isoperimetric inequality. It's why bubbles are round, why planets are round, and why heat-retaining objects (igloos, animals in cold climates) tend toward spherical shapes. A cube of the same volume as a sphere has about 24% more surface area.
Surface area shows up everywhere you have an interface between an object and its environment. Heat transfer through a wall, drag on an aircraft, paint and cladding costs, chemical reactions at a catalyst, cell biology, radiator efficiency, sandpaper grain — all depend on the area exposed. It's one of the most practically useful geometric quantities.