Calculate the surface area of common 3D shapes instantly with formulas, diagrams, worked examples and step-by-step solutions.
Surface area is the total area of every face and curved surface of a 3D object, measured in square units. Add up each surface: a cube is 6s² because it has six identical faces; a cylinder is 2πr² + 2πrh — two circular ends plus the curved side unrolled into a rectangle; a sphere is 4πr². Pick a shape below, enter its dimensions, and the calculator returns the total surface area broken down face by face, plus the lateral area, the volume, the surface-area-to-volume ratio and the working.
Choose a shape:
Enter the dimensions — results update as you type. Every measurement must use the same unit.
| Unit | Surface area |
|---|
For flat shapes use the area calculator; for capacity use the volume calculator.
| Component | Formula | Value |
|---|
Surface area is the total area of all the outer surfaces of a three-dimensional object. If you could peel the object and lay every face out flat, the surface area is how much flat material you would have. Because it is still an area, it is always measured in square units — square millimetres, square metres, square feet — never cubic ones.
That distinction matters because surface area answers a different question from volume. Volume tells you how much fits inside a shape; surface area tells you how much material is needed to cover it. Paint, plaster, cladding, wrapping, plating and insulation are all surface-area problems. Concrete, water, grain and air are volume problems.
The flattened arrangement of faces is called the net of a shape, and it is the most reliable way to approach any unfamiliar solid: identify every surface, work out the area of each, then add them. Every formula on this page is just that process already done for you.
These three get confused constantly, usually because two of them share the same units. This is the whole distinction in one table.
| Measurement | Dimension | Unit | Everyday example | What it answers |
|---|---|---|---|---|
| Area | 2D — a flat shape | m² | A floor, a wall, a block of land | How much surface to cover, flat |
| Surface area | 3D — the exterior | m² | All the walls and roof of a house | How much material to wrap or coat it |
| Volume | 3D — the interior | m³ | Water inside a tank | How much fits inside |
Area and surface area both use square units because both measure surface — the difference is whether that surface is flat or wrapped around a solid. Volume is the odd one out with cubic units, because it measures three dimensions multiplied together. A useful check: if you are buying something sold by the square metre, it is an area or surface-area problem; by the cubic metre or the litre, it is volume.
Total surface area (TSA) counts every surface. Lateral surface area (LSA) counts only the sides, excluding the base or bases. Which one you want depends entirely on the job.
| Shape | Total surface area | Lateral surface area | When lateral is the one you need |
|---|---|---|---|
| Cube | 6s² | 4s² | Painting four walls but not the floor or ceiling |
| Cuboid | 2(lw + lh + wh) | 2h(l + w) | Wallpapering a room |
| Cylinder | 2πr² + 2πrh | 2πrh | The label around a tin, or lagging a pipe |
| Cone | πr² + πrl | πrl | Fabric for a conical roof or a party hat |
| Square pyramid | b² + 2bl | 2bl | Roof sheeting on a pyramid roof |
| Prism | 2A + Ph | Ph (perimeter × length) | Cladding a prism-shaped duct |
| Sphere | 4πr² | Not applicable | A sphere has no base to exclude |
TSA = 6s². A cube has six identical square faces, so calculate one face and multiply by six. A cube of side 5 has faces of 25, giving 150 square units. Lateral area is 4s² = 100, leaving out the top and bottom.
TSA = 2(lw + lh + wh). Three pairs of identical rectangles: top and bottom, front and back, and the two ends. For 8 × 5 × 4 the pairs are 40, 32 and 20, so TSA = 2 × 92 = 184 square units. The lateral area, 2h(l + w) = 104, is exactly what you need for the four walls of a room.
TSA = 2πr² + 2πrh. The two circular ends contribute 2πr². The curved side is the elegant part: unroll it and it becomes a plain rectangle, 2πr wide (the circumference) by h tall, so its area is 2πrh. For r = 5 and h = 10 that gives 157.08 + 314.16 = 471.24 square units. The formula is often written more compactly as 2πr(r + h).
TSA = πr² + πrl, where l is the slant height, not the vertical height. Find it first with l = √(r² + h²) — Pythagoras again. For r = 4 and h = 9, l = √97 ≈ 9.849, so the curved surface is πrl ≈ 123.76 and the base is πr² ≈ 50.27, totalling about 174.03 square units. Substituting h for l here is the single most common cone mistake.
TSA = 4πr². A sphere has no edges, faces or base — one continuous curved surface. The result is exactly four times the area of a circle with the same radius, which Archimedes proved over two thousand years ago. For r = 10 the surface area is 4π × 100 ≈ 1,256.64 square units.
TSA = 3πr². Half a sphere's curved surface is 2πr², and slicing it exposes a flat circular face of πr², so a solid hemisphere totals 3πr². For r = 8 that is 603.19 square units. If the shape is an open dome — a bowl or a roof — there is no flat face and only the curved 2πr² ≈ 402.12 counts. Deciding whether the flat face is included is what most hemisphere errors come down to.
TSA = 2 × base area + perimeter × length. Any prism follows this pattern: two identical ends plus a wrap-around lateral surface. For a triangular prism with base sides 3, 4 and 5 and a length of 10, Heron's formula gives a base area of 6, so TSA = 2(6) + 12 × 10 = 132 square units. The same logic covers hexagonal, pentagonal and irregular prisms — only the base area and perimeter change.
TSA = b² + 2bl for a square pyramid, where l = √(h² + (b/2)²) is the slant height of a triangular face. Each face is ½ × b × l and there are four, giving 2bl in total. With b = 6 and h = 8, l = √(64 + 9) ≈ 8.544, so the lateral area is about 102.53 and the base 36, totalling 138.53 square units. Note the lateral edge is longer again at √(h² + 2(b/2)²) ≈ 9.06 — useful for cutting hip rafters, but not the value that goes in the area formula.
| Field | Why surface area matters | Example |
|---|---|---|
| Construction | Cladding, render, insulation, formwork and waterproofing quantities | Wrapping a 3 m × 6 m silo takes about 63.6 m² of sheeting |
| Painting | Coverage rates are quoted per square metre | 40 m² at 10 m²/L needs 4 L per coat |
| Architecture | Facade area, glazing ratios, thermal performance | Heat loss scales with envelope area, not floor area |
| Manufacturing | Sheet metal nesting, plating, powder coating, anodising | Plating cost is charged by surface area |
| Packaging | Cardboard and film per unit, print area | A cube uses more material per litre than a sphere |
| Engineering | Heat exchange, drag, friction, corrosion exposure | Radiator fins exist purely to add surface area |
| Biology | Absorption and exchange rates | Villi and alveoli multiply surface area enormously |
| Chemistry | Reaction rate depends on exposed area | Powdered reactants react far faster than lumps |
Almost every building product is sold by the square metre or by a coverage rate, so a surface area figure is what turns a design into an order. These are the conversions that come up most often.
| Product | What you need | Typical rate | Worked example |
|---|---|---|---|
| Paint | Wall area minus openings, then coats | Around 10–16 m² per litre per coat | 40 m² at 10 m²/L = 4 L per coat, so 8 L for two coats |
| Tiles | Surface area ÷ area per tile | A 300 × 600 mm tile covers 0.18 m² | 12 m² ÷ 0.18 = 67 tiles, order about 74 with 10% waste |
| Insulation batts | Wall or ceiling area | Sold per pack with an m² coverage printed on it | A 96 m² ceiling with packs covering 8 m² needs 12 packs |
| Sheet metal / cladding | Surface area plus laps | Sheets quoted in m², laps add 5–10% | A silo at 63.6 m² → order roughly 70 m² |
| Wrapping paper | Total surface area of the box | Add roughly 15% for folds and overlap | A 30 × 20 × 10 cm box is 2,200 cm², so allow about 2,530 cm² |
| Swimming pool lining | Floor plus all four walls | Liner and tiling both priced per m² | An 8 × 4 m pool 1.5 m deep: 32 + 2(1.5)(8+4) = 68 m² |
| Plasterboard | Wall and ceiling area ÷ sheet area | A 2400 × 1200 mm sheet covers 2.88 m² | 45.6 m² ÷ 2.88 ≈ 16 sheets, order 18 |
| Powder coating / plating | Total surface area of the part | Charged per m² of coated surface | Coating cost scales directly with surface area, not weight |
Surface area and volume grow at different rates, and that single fact explains a surprising amount of the physical world. Double every dimension of an object and its surface area increases fourfold while its volume increases eightfold. So the surface-area-to-volume ratio falls as things get bigger.
| Cube side | Surface area | Volume | SA : V ratio |
|---|---|---|---|
| 1 | 6 | 1 | 6.00 |
| 2 | 24 | 8 | 3.00 |
| 5 | 150 | 125 | 1.20 |
| 10 | 600 | 1,000 | 0.60 |
Consequences turn up everywhere. Small animals lose body heat quickly because they have a lot of surface for their volume, which is why they eat proportionally more. Ice cubes melt faster than a single large block of the same total mass. Crushed ore dissolves faster than rock. And of all shapes enclosing a given volume, the sphere has the least surface area, which is why bubbles, droplets and pressure vessels are round — and why a cube-shaped package uses more cardboard per litre than a rounder one. The calculator reports this ratio for every shape.
Measuring flat land came first; measuring the skin of a solid took much longer. Egyptian and Babylonian scribes could already calculate areas and some volumes by around 1800 BCE, but they worked from practical recipes rather than proofs.
Around 300 BCE Euclid's Elements put solid geometry on a deductive footing, defining prisms, pyramids, cylinders, cones and spheres and proving relationships between them. Half a century later Archimedes solved the hardest case of all. Using the method of exhaustion — bounding a curved surface between polygons and letting the number of sides grow — he proved that a sphere's surface area is exactly 4πr², four times the area of its own great circle, and that a sphere inscribed in a cylinder has two thirds of the cylinder's surface area and two thirds of its volume. He considered this his finest result and reportedly asked for the sphere-and-cylinder diagram to be carved on his tomb.
Those methods anticipated integral calculus by nearly two millennia, and calculus later generalised them to any curved surface. Today the same formulas drive material take-offs, heat-transfer and drag calculations, corrosion allowances and coating costs, standardised through SI units so a square metre means the same thing everywhere.
| Mistake | Why it goes wrong | How to avoid it |
|---|---|---|
| Using vertical height instead of slant height | Cones and pyramids need the sloping length, which is always longer | Calculate l = √(r² + h²) first, every time |
| Giving the answer in cubic units | Surface area is an area, so units are squared | Check the unit: m², never m³ |
| Forgetting a face | The base or the second end gets left out | Sketch the net and count the surfaces |
| Counting a face that is not there | An open tank, pipe or dome has no lid | Decide which surfaces are actually being covered |
| Using diameter as radius | Every πr² term becomes four times too big | Halve the diameter first |
| Mixing units | Metres with centimetres gives a meaningless figure | Convert everything before calculating |
| Confusing surface area with volume | They answer different questions | Covering is area; filling is volume |
| Doubling dimensions and doubling the answer | Area scales with the square of length | Double the size and surface area quadruples |
| Ignoring waste and openings | Real jobs have laps, offcuts, doors and windows | Subtract openings, then add 5–10% for waste |
TSA = 6s² | LSA = 4s² | V = s³TSA = 2(lw + lh + wh) | LSA = 2h(l + w) | V = lwhTSA = 2πr(r + h) | LSA = 2πrh | V = πr²hTSA = πr(r + l), l = √(r² + h²) | V = ⅓πr²hTSA = 4πr² | V = ⁴⁄₃πr³TSA = 3πr², dome only 2πr² | V = ⅔πr³TSA = b² + 2bl, l = √(h² + (b/2)²) | V = ⅓b²hTSA = 2A + Ph | V = AhTSA = 4πr² + 2πrh | V = πr²h + ⁴⁄₃πr³TSA = 4π²Rr | V = 2π²Rr²TSA = π(r₁+r₂)l + πr₁² + πr₂², l = √(h² + (r₂−r₁)²)×k on lengths → ×k² on surface area, ×k³ on volume
1) 6 × 16 = 96 cm² 2) 2(24 + 18 + 12) = 108 cm² 3) 4π × 25 ≈ 314.16 cm² 4) 4 × 16 = 64 cm² 5) Square units, cm²
1) 2π(9) + 2π(3)(10) = 56.55 + 188.50 ≈ 245.04 cm² 2) l = √(36 + 64) = 10 cm; TSA = π(6)(6 + 10) = 96π ≈ 301.59 cm² 3) Solid: 3π(64) ≈ 603.19 cm²; open dome: 2π(64) ≈ 402.12 cm² 4) l = √(144 + 25) = 13 m, so TSA = 100 + 2(10)(13) = 360 m² 5) Walls = 2 × 2.4 × 9 = 43.2 m²; two coats = 86.4 m² ÷ 10 ≈ 8.64 L, so buy 10 L 6) Side 2: 24/8 = 3.0. Side 6: 216/216 = 1.0. The smaller cube has three times the surface area per unit of volume 7) s = 12, base area = √(12 × 6 × 4 × 2) = 24; TSA = 2(24) + 24 × 15 = 408 square units 8) The squat one, closer to the proportions where height equals the diameter — that shape minimises surface area for a fixed cylindrical volume, which is why tins are shaped the way they are
What is surface area?
Surface area is the total area of all the outer surfaces of a three-dimensional object — every flat face plus every curved surface. It is measured in square units such as cm² or m², because it is still an area even though the object is solid. If you flattened the object into its net, the surface area is how much flat material you would have.
How do you calculate surface area?
Identify every surface, calculate the area of each, then add them together. For a cuboid that means three pairs of rectangles: 2(lw + lh + wh). For a cylinder it means two circles plus the curved side unrolled into a rectangle: 2πr² + 2πrh. Curved shapes use established formulas — 4πr² for a sphere, πr² + πrl for a cone. Sketching the net is the most reliable approach for any unfamiliar solid.
What is the surface area formula?
Each shape has its own. The most used are 6s² for a cube, 2(lw + lh + wh) for a cuboid, 2πr² + 2πrh for a cylinder, πr² + πrl for a cone, 4πr² for a sphere, 3πr² for a solid hemisphere, and b² + 2bl for a square pyramid. Any prism follows the general pattern: twice the base area plus the base perimeter times the length.
How do you find the surface area of a cube?
Multiply one face by six: TSA = 6s². A cube has six identical square faces, so a cube of side 5 has faces of 25 and a total surface area of 150 square units. If you only need the four sides — painting walls but not the floor or ceiling — the lateral surface area is 4s² = 100.
How do you calculate the surface area of a cylinder?
TSA = 2πr² + 2πrh, often written 2πr(r + h). The 2πr² covers the two circular ends and the 2πrh covers the curved side: unroll it and it becomes a rectangle 2πr wide by h tall. For r = 3 and h = 10: 2π(9) + 2π(30) ≈ 56.55 + 188.50 ≈ 245.04 square units. For an open pipe use only the curved 2πrh.
How do you calculate the surface area of a sphere?
TSA = 4πr². A sphere has one continuous curved surface with no faces or base, so there is no lateral area to separate out. A radius of 10 gives 4π × 100 ≈ 1,256.64 square units. The result is exactly four times the area of a circle of the same radius, a relationship Archimedes proved in the third century BCE.
How do you calculate the surface area of a cone?
TSA = πr² + πrl, where l is the slant height — the sloping distance from the base edge to the tip. Find it first with l = √(r² + h²). For r = 4 and h = 9, l = √97 ≈ 9.849, so the curved surface is πrl ≈ 123.76, the base is πr² ≈ 50.27, and the total is about 174.03 square units. Using the vertical height in place of l is the most common cone error.
What is total surface area?
Total surface area counts every surface of the solid, including the base and any second end. It is what you need when the whole object is being coated, wrapped or plated — dipping a component, painting a free-standing sculpture, or wrapping a parcel.
What is lateral surface area?
Lateral surface area counts only the sides, excluding the base or bases. For a cube it is 4s² rather than 6s²; for a cylinder 2πrh rather than 2πrh + 2πr². It is the figure you want for wallpapering a room, lagging a pipe, printing a label around a tin, or cladding the walls of a tank whose base sits on the ground.
What is the difference between area and surface area?
Area applies to a flat two-dimensional shape — a rectangle, a circle, a triangle. Surface area applies to a three-dimensional object and totals the areas of all its surfaces. Both use square units, so the difference is what is being measured, not the units. Use the area calculator for flat shapes and this page for solids.
How is surface area different from volume?
Surface area measures the covering; volume measures the contents. Surface area uses square units, volume uses cubic units. Paint, cladding and wrapping are surface-area problems; concrete, water and grain are volume problems. They also scale differently: double the dimensions and surface area quadruples while volume grows eightfold.
What is the surface-area-to-volume ratio?
It is surface area divided by volume, and it falls as objects get bigger. A cube of side 1 has a ratio of 6; at side 10 it is only 0.6. This explains why small animals lose heat quickly and eat proportionally more, why crushed ore dissolves faster than rock, why ice cubes melt faster than one large block, and why cells stay microscopic. The calculator reports the ratio for every shape.
Why does a sphere have the smallest surface area for a given volume?
Because a sphere is the only shape with no corners, edges or flat faces — every point on its surface sits the same distance from the centre, so none of the surface is "wasted" reaching out further than it needs to. Any bump, corner or elongation adds surface without adding a matching amount of volume. Compare equal volumes of 1,000 cm³: a sphere needs about 483.6 cm² of surface, a cube needs 600 cm², and a long thin box needs far more again. This is why bubbles and droplets pull themselves into spheres, since surface tension minimises surface energy; why cells, planets and pressure vessels are round; and why a spherical tank uses the least steel for a given capacity. The reverse also holds: when you want maximum surface area, as in a radiator, a heat sink or the lining of a lung, you move as far from a sphere as possible with fins, folds and branches.
How do builders use surface area?
For quantities and costs. Cladding, render, plasterboard, insulation, waterproof membrane and formwork are all bought and priced by the square metre, so the take-off starts with surface area. Cladding a 3 m diameter, 6 m tall silo needs about 63.6 m² including the roof, and a practical order adds roughly 10% for laps and offcuts.
How do painters calculate surface area?
Work out the wall area, subtract the openings, then divide by the coverage rate on the tin. For a 4 m × 3 m room with 2.4 m ceilings the walls are 2 × 2.4 × 7 = 33.6 m², plus 12 m² of ceiling. Taking off a door and two windows leaves roughly 40.7 m², so at 10 m² per litre that is about 4.1 L per coat — buy 10 L for two coats and allow extra for a porous or dark surface.
How do engineers calculate surface area?
Usually because a rate depends on it. Heat transfer, drag, friction, corrosion exposure and plating cost are all proportional to surface area, which is why radiators and heat sinks are built as fins — the entire purpose is to add area without adding bulk. Engineers also use it in reverse, minimising surface area to reduce material cost or heat loss for a required volume.
How do architects calculate surface area?
Building envelope area drives thermal performance, so heat loss and solar gain calculations depend on wall, roof and glazing areas rather than floor area. Facade material quantities, glazing ratios and energy compliance assessments all start from surface area, taken from the coordinate model rather than scaled off a drawing.
What units are used for surface area?
Square units: square millimetres, square centimetres and square metres in metric, or square inches and square feet in imperial. Australia uses the metric system, with mm² common in manufacturing and m² in construction. Conversion factors are the square of the length factors, so 1 m² = 10,000 cm², and this calculator converts automatically.
Can this calculator solve all 3D shapes?
It covers eleven: sphere, cube, cuboid, cylinder, cone, square pyramid, triangular prism, capsule, hemisphere, torus and frustum. For shapes not listed, break the object into these components, calculate each part, and add — but leave out any surfaces that become internal joins, since those are no longer exposed.
What is the net of a shape?
A net is the shape unfolded into a flat pattern of all its faces. Unfolding a cube gives six squares; unfolding a cylinder gives two circles and a rectangle. Nets are the most reliable way to work out surface area, because every face becomes visible and nothing gets counted twice or missed. Packaging designers work directly with nets when laying out cardboard.
Why does a cone use slant height instead of vertical height?
Because the curved surface follows the slope, not the axis. Unroll a cone's curved surface and it becomes a sector of a circle whose radius is the slant height, so l is the dimension that governs the area. The slant height is always longer than the vertical height, being the hypotenuse of the triangle formed by r and h — so using h instead always understates the answer.
How do students calculate surface area?
The method that works under exam pressure is to sketch the net, label every face with its dimensions, calculate each area, then add. Write the formula down before substituting numbers, calculate slant heights as a separate first step, and check the final unit is squared. Working through the practice questions on this page with the calculator open is a good way to confirm each step.
Why is surface area important?
Because so many real quantities and rates depend on it: material for cladding and packaging, paint coverage, heat transfer, drag, corrosion, reaction speed and biological absorption. It is also the number that turns a design into a cost, since coatings and sheet materials are priced per square metre.
What are common surface area mistakes?
Using vertical height where slant height is required in cones and pyramids; giving the answer in cubic units; forgetting a face, or counting one that does not exist on an open tank or pipe; using the diameter as the radius; mixing units; confusing surface area with volume; and assuming that doubling the dimensions doubles the surface area when it actually quadruples it.
Does this surface area calculator show the formula and steps?
Yes. For every shape it displays the formula, a face-by-face breakdown with each component's own formula, the total and lateral surface areas, the volume, the surface-area-to-volume ratio, and the working line by line. It also draws a labelled diagram of the shape, converts the result into mm², cm², m², in² and ft², updates as you type, and lets you copy everything with one tap.
A one-page reference with surface area and volume for all eleven shapes plus the scaling rules. Use the button to print it or save it as a PDF — the rest of the page is hidden from the printout.
MegaCalcOnline.com · 3D shapes: total surface area, lateral surface area and volume
| Shape | Total surface area | Lateral / notes | Volume |
|---|---|---|---|
| Cube | 6s² | LSA = 4s² | s³ |
| Cuboid | 2(lw + lh + wh) | LSA = 2h(l + w) | lwh |
| Cylinder | 2πr(r + h) | LSA = 2πrh | πr²h |
| Cone | πr(r + l) | l = √(r² + h²); LSA = πrl | ⅓πr²h |
| Sphere | 4πr² | No base to exclude | ⁴⁄₃πr³ |
| Hemisphere | 3πr² | Open dome = 2πr² | ⅔πr³ |
| Square pyramid | b² + 2bl | l = √(h² + (b/2)²) | ⅓b²h |
| Prism (any) | 2A + Ph | A = base area, P = base perimeter | Ah |
| Capsule | 4πr² + 2πrh | Sphere + cylinder side | πr²h + ⁴⁄₃πr³ |
| Torus | 4π²Rr | Requires r < R | 2π²Rr² |
| Frustum | π(r₁+r₂)l + πr₁² + πr₂² | l = √(h² + (r₂−r₁)²) | ⅓πh(r₁² + r₁r₂ + r₂²) |
| Rules & conversions | Value |
|---|---|
| Scaling | ×k on lengths → ×k² on surface area, ×k³ on volume |
| Surface area units | Always squared: mm², cm², m² |
| 1 m² | = 10,000 cm² = 1,000,000 mm² ≈ 10.7639 ft² |
| SA : V ratio | Falls as objects get larger |
| Least surface area for a volume | The sphere |
| Paint estimate | area ÷ coverage (m²/L) × number of coats |
| Practical allowance | Add 5–10% for laps, offcuts and waste |
Educational use only. Cones and pyramids use slant height, never vertical height.