Surface Area Calculator — Cube, Box, Sphere, Cylinder, Cone & More
Pick from ten solids, type each measurement in its own unit — centimetres, inches, metres and more — and get the total surface area, broken into lateral area, caps and faces, plus the volume, all from the exact geometry formulas.
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Surface area
- Ten solids — cylinder, cube, box, sphere, cone, capsule, frustum, ellipsoid, spherical cap and square pyramid — and each shape asks only for the measurements its formula actually needs.
- The total is always the sum of its parts: at the defaults the cylinder’s lateral 314.16 + caps 157.08 = 471.24 cm², with the volume (785.40 cm³) shown alongside as a bonus.
- Lateral area is only the curved or side walls; total surface area adds every flat face — painting a tin can’s outside label area is lateral, working out the sheet metal for the whole can is total.
Meet the ten solids
The diagram follows the shape you pick above — every measurement you type is the one the formula uses.
Keep exploring
Three companion tools for flat shapes and spaces.
How it works
Three steps from a solid to its skin.
Pick your solid
Ten solids — cylinder, cube, box, sphere, cone, capsule, frustum, ellipsoid, spherical cap and square pyramid — each shape asks only for the measurements its formula actually needs.
Type the measurements
Radius, height, edge — each in its own unit: centimetres, inches, metres and more. Results update as you type, with no button to press.
Read the breakdown
The total surface area, split into the parts that build it — lateral wall, caps, faces — plus the volume as a bonus.
The math behind the areas
Surface area is the total area of a solid’s outer skin — imagine unwrapping the shape and laying every face flat. Each classic solid has an exact formula built from π and its own measurements. A cylinder’s curved wall unrolls into a rectangle whose width is the circumference, which is why its lateral area is 2πrh; a cone needs its slant height √(r² + h²), the distance down the slope, before its curved area πr × slant can be found.
Worked examples — the ten shapes at this page’s starting sizes:
| Shape | Numbers in | Working | Total surface area | Volume |
|---|---|---|---|---|
| Cylinder | r = 5, h = 10 | lateral 314.16 + caps 157.08 | 471.24 cm² | 785.40 cm³ |
| Cube | s = 5 | 6 × 5² | 150.00 cm² | 125.00 cm³ |
| Box | 4 × 5 × 6 | 2 × (20 + 24 + 30) | 148.00 cm² | 120.00 cm³ |
| Sphere | r = 5 | 4π × 5² | 314.16 cm² | 523.60 cm³ |
| Cone | r = 3, h = 4 | slant 5 → π × 3 × (3 + 5) | 75.40 cm² | 37.70 cm³ |
| Capsule | r = 3, length = 6 | wall 113.10 + sphere 113.10 | 226.19 cm² | 282.74 cm³ |
| Frustum | R = 5, r = 3, h = 4 | slant 4.47 → lateral 112.40 + bases 106.81 | 219.21 cm² | 205.25 cm³ |
| Ellipsoid | a = 3, b = 2, c = 1 | Knud Thomsen approximation | ≈ 48.97 cm² | 25.13 cm³ |
| Spherical cap | R = 10, h = 4 | curved 251.33 + base 201.06 | 452.39 cm² | 435.63 cm³ |
| Square pyramid | s = 6, h = 4 | slant 5 → base 36.00 + faces 60.00 | 96.00 cm² | 48.00 cm³ |
A neat coincidence in the table: a cylinder with r = 5, h = 10 has exactly the same lateral area (314.16 cm²) as the total surface of a sphere with r = 5 — because 2πrh and 4πr² meet when h = 2r.
Go deeper
Lateral vs total, and why the skin of a shape runs the show.
Two numbers get quoted for the same solid, and mixing them up is the classic homework error:
- Lateral area: only the curved or side walls — for a cylinder, the 2πrh rectangle you get by unrolling it, with no caps.
- Total surface area: lateral area plus every flat face — the cylinder adds its two circular caps (2πr²), the cone adds its base (πr²).
- Which one you need: painting a tin can’s outside label area is lateral; working out the sheet metal to make the whole can is total. A cube and a box have no curved part, so lateral is not usually quoted for them.
- Open shapes: an open-top box or a tank without a lid simply drops that face from the total — subtract one face area (for a cylinder, one cap) from the figure above.
Surface area quietly controls a lot of everyday engineering and biology:
- Paint and packaging: coverage is sold by area — a shape’s total surface area is exactly the paint, wrap or sheet material it needs, before waste.
- Heat and cooling: radiators, cooling fins and elephants’ ears all grow surface area to shed heat faster; a small animal loses heat quickly for the same reason.
- Chemistry and biology: reactions happen at surfaces — crushed reactants react faster, and your lungs pack roughly a tennis court of surface into your chest for gas exchange.
- The square–cube law: double a shape’s size and its area grows ×4 while its volume grows ×8 — which is why giant animals need special engineering and why small cubes dissolve faster than big ones.
Surface area, area, and volume: three different questions
Same shape, three different measurements — and three different units.
Area measures one face. Surface area measures all the faces added together. Volume measures the space inside. A box can have a large surface area and a small volume (a flat, wide tray) or the reverse (a near-cube), so the three numbers answer different questions and must never be swapped. Units keep them apart: area and surface area are always in square units (cm², m², ft²), while volume is in cubic units.
Lateral area counts only the sides — the curved wall of a cylinder, or the four walls of a box — and leaves the top and bottom out. Total surface area adds those faces back in. The distinction matters most for open shapes: a tank with no lid, a room with no ceiling to paint, or a cone with no base all use lateral area plus only the faces that exist. If the calculator on this page returns a total and your object is open, subtract the missing face by hand and say so in your working.
Land, floor, and paint: when the textbook formula is not the answer
Irregular plots, real rooms, and coverage rates need one extra step each.
Three everyday uses stretch the textbook formulas. Land area for an irregular plot is rarely one rectangle: split the plot into rectangles and triangles, work out each piece, and add them — or, for a plot with a curved boundary, measure the longest length and the average width and treat the result as an estimate, not a survey. Floor area for ordering flooring needs the same split-and-add approach around alcoves, plus the wastage the material demands; the calculator gives the geometric area, and waste is your addition. Paint needs one more division: total surface area divided by the coverage on the tin (often around 13 m² per litre per coat), then a second coat and roughly 10% for waste and rough surfaces on top.
The habit that prevents most errors is naming the faces before calculating. Write down which faces exist, which are open, and which two faces share an edge, then check the total against a rough estimate: a cube with 5 cm sides has six faces of 25 cm², so 150 cm². If your answer is far from a check like that, a face has been missed or counted twice.
Frequently asked questions
Quick answers about surface area.
How do I calculate the surface area of a cylinder?
Use 2πr(r + h): the curved wall (2πrh) plus the two circular caps (2πr²). With r = 5 cm and h = 10 cm, the wall is 314.16 cm², the caps add 157.08 cm², and the total surface area is 471.24 cm².
What is the surface area of a sphere with radius 5 cm?
4πr² = 4 × π × 25 = 314.16 cm². A sphere has no flat faces, so its total surface area is exactly four times the area of its great circle (πr² = 78.54 cm²).
How do you find the surface area of a cone?
First find the slant height with Pythagoras: slant = √(r² + h²). Then the total is πr(r + slant). For r = 3 cm and h = 4 cm the slant is exactly 5 cm, giving π × 3 × 8 = 75.40 cm² — 47.12 cm² of curved surface plus a 28.27 cm² base.
What is the difference between lateral and total surface area?
Lateral area counts only the side or curved walls; total surface area adds the flat faces too. A cylinder’s lateral area is 2πrh, and its total adds the two caps. If your shape is open — a box with no lid, a tank with no top — subtract the missing face from the total.
Why is surface area measured in square centimetres?
Area always carries squared units: you are multiplying a length by a length (cm × cm = cm²). Measure in metres instead and the answer comes out in m² — divide the cm² figure by 10,000 to convert. Volume is one step further: cm × cm × cm = cm³, which is why the volume figures on this page convert to litres by dividing by 1,000.
How much paint do I need to cover a shape?
Divide the total surface area by the paint’s coverage. A typical tin covers about 13 m² per litre per coat: a cylinder with 471.24 cm² (0.047 m²) of surface needs only about 4 ml per coat — but real jobs add a second coat and roughly 10% for waste, drips and rough surfaces, so the area here is your starting point, not the whole shopping list.
How do you find the surface area of an ellipsoid?
There is no exact formula in everyday functions, so this calculator uses the Knud Thomsen approximation with p = 1.6075, which is accurate to within about ±1.07% — that is why ellipsoid results carry a ≈ sign. For semi-axes a = 3, b = 2 and c = 1 it gives ≈ 48.97 cm². The ellipsoid’s volume, (4/3)πabc, is exact.
Does surface area include the bottom of an open box?
No — not unless the bottom exists. Total surface area counts every face the solid actually has, so an open-top box counts the four sides and the base, but not a lid it does not have. Work out the total first, then subtract the missing face, or add the faces that exist one by one. Say which you did, because a reader who assumes a closed box will get a different answer.
How do I work out surface area from a net?
Add the area of every shape in the net, face by face. A net lays a solid flat, so each rectangle, triangle or circle in it is one face of the solid. Find each face’s area with its own formula, add them all, and check the count: a cube’s net has six squares, a square pyramid’s has one square and four triangles. If your count of faces does not match the solid, a face is missing or one has been counted twice.
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Disclaimer
This calculator is for education and estimation. It uses perfect geometric solids; real objects have seams, thickness, rounded edges and surface texture that change the true area. For fabrication or ordering materials, confirm with your own measurements and allow for waste.