Geometry
Surface Area Calculator
Find the surface area of nine common solids — sphere, cube, box, cylinder, cone, pyramid, prism, hemisphere, and capsule — straight from their dimensions. Pick a shape, type the measurements, and watch the total, the piece-by-piece breakdown, the volume, and the surface-area-to-volume ratio update live.
Choose a solid
Cylinder dimensions
2 inputsSA = 2πr² + 2πrh
Total surface area (cm2)
188.4956
Breakdown
- 2 circular ends (2πr²)56.5487 cm2
- Curved side (2πrh)131.9469 cm2
Lateral area
131.9469 cm²
Volume
197.9203 cm³
SA : volume
0.952 : 1
Live — updates as you type. SA in cm², volume in cm³.
No sign-up · Free
The basics
What is surface area?
Surface area is the total area of every outer face of a 3D object — the amount of material it would take to wrap, paint, or skin it completely. Because area multiplies two lengths, the answer is always in square units. The trick to every formula is the same: split the solid into faces you can measure, then add them up.
SA = Σ (area of each face)
Add up the faces
Every surface-area formula is a sum of simpler areas — circles, rectangles, and triangles. A cube is six squares; a box is three pairs of rectangles; a cylinder is two circles plus one rolled-up rectangle.
cube: 6 equal squares → 6a²
total = lateral + bases
Lateral plus bases
For solids with a top and bottom it helps to separate the lateral area — the sides — from the base area. The calculator shows that split so you can check each piece.
cylinder: 2πrh (side) + 2πr² (ends)
Method
How to find surface area by hand
Three steps work for any solid, no matter how many faces it has.
- 1
Identify the faces
List every surface of the solid and its shape. A cylinder, for example, has two circular ends and one curved side that unrolls into a rectangle.
cylinder → 2 circles + 1 curved wall - 2
Find each area
Apply the right area formula to each face. Circles use πr², rectangles use length × width, triangles use ½ × base × height.
ends: 2πr² side: 2πrh - 3
Add them together
Sum the face areas. The single number you get, in square units, is the total surface area.
SA = 2πr² + 2πrh
Worked examples
See it across solids
The same add-the-faces method, shown end to end for five common solids. Switch tabs to watch each formula fill in from real dimensions.
Formula SA = 4πr²
Given r = 5
Substitute and evaluate each piece:
• Surface (4πr²) = 314.1593
SA = 314.1593 units²
Reference
The surface-area formulas
One card per solid the calculator supports. Every result comes out in square units.
Sphere
SA = 4πr²
four times the area of a great circle.
Cube
SA = 6a²
six identical square faces.
Rectangular box
SA = 2(lw + lh + wh)
twice the sum of the three distinct face areas.
Cylinder
SA = 2πr² + 2πrh
two circular ends plus the curved side.
Cone
SA = πr² + πrl
circular base plus the slanted side, with slant l = √(r² + h²).
Square pyramid
SA = a² + 2a·l
square base plus four triangles, with slant l = √(h² + (a/2)²).
Triangular prism
SA = 2·(½bh) + (a + b + c)·L
two triangular ends plus three rectangular sides.
Hemisphere
SA = 3πr²
the curved dome (2πr²) plus the flat circular base (πr²).
Capsule
SA = 4πr² + 2πrh
a full sphere of caps plus a cylindrical middle of height h.
Quick chart
Surface area by size, at a glance
The most-searched lookup: the surface area of a sphere, cube, or hemisphere for whole-number sizes. Switch the solid and read off the value — no typing needed.
Sphere surface area by radius (SA = 4πr²)
radius = 1
12.5664
radius = 2
50.2655
radius = 3
113.0973
radius = 4
201.0619
radius = 5
314.1593
radius = 6
452.3893
radius = 7
615.7522
radius = 8
804.2477
radius = 9
1,017.88
radius = 10
1,256.64
radius = 12
1,809.56
radius = 15
2,827.43
radius = 20
5,026.55
radius = 25
7,853.98
radius = 50
31,415.93
radius = 100
125,663.71
Where it shows up
Why surface area matters
Surface area decides how much material you buy and how fast things heat, cool, react, or dry.
paint = SA ÷ coverage
Building & DIY
Paint, tiling, wrapping, and sheet-metal jobs all price by area, so the surface area sets the material you need to cover the object.
heat loss ∝ SA
Engineering & cooking
Heat flows through surfaces. A thin, high-surface shape cools or cooks faster than a compact one of the same volume — why radiators have fins.
SA : V ratio
Biology & chemistry
Cells, lungs, and catalysts maximise surface area per unit volume to speed up exchange and reactions. The ratio is the key design number.
Going deeper
The surface-area-to-volume ratio
Surface area and volume scale differently — and that gap explains a surprising amount of the physical world.
SA ∝ r², V ∝ r³
Bigger means smoother
Double an object's size and its surface area grows fourfold while its volume grows eightfold. So the larger something gets, the less surface it has per unit of volume.
double r → SA ×4, V ×8
SA : V drops as size rises
Why small things differ
A small object has lots of surface relative to its insides, so it cools, dries, and reacts quickly. The tool reports this ratio for every shape you enter.
sphere r = 1 → 3 : 1; r = 10 → 0.3 : 1
Worth knowing
Rules and common mistakes
A handful of habits that keep surface-area answers correct.
units²
Always square the unit
Surface area is a length times a length, so the unit is squared. Inputs in centimetres give an answer in square centimetres — never plain centimetres.
d = 2r
Diameter is not radius
The most common error. If a problem gives the diameter, halve it before using any formula that takes the radius r.
l = √(r² + h²)
Slant height, not height
Cones and pyramids use the slant height for the sloped faces, which is longer than the vertical height. Compute it first.
convert before squaring
Match the units first
If dimensions are in different units, convert them to one unit before multiplying. Mixing metres and centimetres silently wrecks the answer.
open vs closed
Mind missing faces
An open box, a tube, or a pipe is missing one or both bases. Drop those terms — don't use the closed-solid formula by reflex.
SA(scaled) = k² · SA
Scaling squares the area
Enlarge every dimension by a factor k and the surface area grows by k². Triple a model and it needs nine times the paint.
FAQ
Common questions
Surface area is the total area of every outer face of a three-dimensional object. It tells you how much material it would take to wrap or paint the object completely, and it is always measured in square units such as cm², m², or in².
Use SA = 4πr², where r is the radius. Square the radius, multiply by π (about 3.14159), then multiply by 4. For a radius of 5 that is 4 × π × 25 ≈ 314.16 square units.
Use SA = 2πr² + 2πrh. The first term, 2πr², is the two circular ends; the second, 2πrh, is the curved side wrapped around. Add the two pieces to get the total. For r = 3 and h = 7, that is about 56.55 + 131.95 ≈ 188.50 square units.
Surface area measures the outside skin of a solid in square units, while volume measures the space it holds in cubic units. A balloon's surface area is how much rubber it takes; its volume is how much air fits inside.
Lateral surface area is the area of the sides only, excluding the top and bottom bases. For a cylinder it is the curved wall (2πrh); for a cone it is the slanted side (πrl). Total surface area adds the bases back on.
Use SA = πr² + πrl, where l is the slant height l = √(r² + h²). The πr² term is the circular base and πrl is the slanted side. You usually compute the slant height first, then plug it in.
Always square units — square centimetres (cm²), square metres (m²), square inches (in²), and so on. If your dimensions are in centimetres, the surface area comes out in square centimetres because area multiplies two lengths.
Yes. A rectangular prism, a rectangular box, and a cuboid all describe the same solid with six rectangular faces. Its surface area is SA = 2(lw + lh + wh), twice the sum of its three distinct face areas.
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Ugo CharlesBuilt and fact-checked against our editorial standards · Last updated June 2026
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