How to Find the Surface Area of a Cylinder
Surface area is the total area of every face that covers a solid, measured in square units. A closed cylinder has three faces: two circular ends and one curved side. Each end has area πr², and the curved side, if you unroll it, is a rectangle as wide as the circle's circumference (2πr) and as tall as the cylinder (h). Adding them gives A = 2πr² + 2πrh.
Worked example: a cylinder with radius 2 m and height 3 m. The two ends contribute 2 × π × 2² = 8π, and the curved side contributes 2 × π × 2 × 3 = 12π, so the total is 20π (about 62.83185307) m². The curved part on its own, the lateral area, is 12π (about 37.69911184) m². Leave π as a symbol when you want an exact answer, or use 3.14159 for a decimal.
Real objects are often open at one end. A cylindrical tank without a lid has only one circular face, so the area that counts is πr² + 2πrh. For the same dimensions that is 16π (about 50.26548246) m². A pipe open at both ends counts only the curved side, 12π (about 37.69911184) m². The calculator has a switch for exactly this, and its face table shows which faces were included.
How to Find the Surface Area of a Cone
A cone has a circular base and a curved side that tapers to a point. The side is not measured by the height but by the slant height ℓ, the distance along the surface from the rim to the tip. By the Pythagorean theorem ℓ = √(r² + h²). The curved area is πrℓ, and with the base the total is A = πr² + πrℓ.
For a cone with radius 3 m and height 4 m, the slant height is √(9 + 16) = 5 m, so the lateral area is π × 3 × 5 = 15π (about 47.1238898) m² and the total area is 24π (about 75.39822369) m². Remove the base, as with a party hat or a funnel cone, and only the lateral 15π (about 47.1238898) m² remains. The calculator works out the slant height for you and shows it in the notes.
Surface Area Formulas for Common Solids
The table lists the total and lateral formulas for the seven shapes the calculator handles, followed by an example from the tool for each.
| Solid | Total area | Lateral area |
|---|---|---|
| Cylinder | 2πr² + 2πrh | 2πrh |
| Cone | πr² + πrℓ | πrℓ |
| Cube | 6a² | 4a² |
| Rectangular prism | 2(lw + lh + wh) | 2h(l + w) |
| Sphere | 4πr² | none (one curved surface) |
| Capsule | 4πr² + 2πrh | 2πrh |
| Spherical cap | 2πRh + πr² | 2πRh |
Examples from the tool: a cube with edge 3 m has total area 54 m² and lateral area 36 m². A box measuring 2 by 3 by 4 m has total area 52 m² and lateral area 40 m². A sphere of radius 3 m has 36π (about 113.0973355) m². A capsule with radius 1 m and a barrel 2 m long has 8π (about 25.13274123) m², made of a full sphere (4π) plus the barrel (4π). A spherical cap, a dome cut from a sphere of radius 2 m with height 2 m, has 12π (about 37.69911184) m² including its flat base.
Total vs. Lateral Surface Area
The two terms answer different questions. Total surface area counts every face of a closed solid, including the top and bottom. Lateral surface area counts only the sides and leaves out the bases. For a cylinder, a cone, a cube or a prism, the lateral area is the total minus the area of the end faces. Which one you need depends on the job: lateral area for a label wrapped around a can or the side walls of a room, total area for painting or wrapping a closed object.
Open containers are a third case. A box-shaped tank with the lid removed counts the four walls and the floor but not the top. For a tank 2 m by 3 m by 4 m high, the closed total is 52 m², the lateral walls are 40 m², and the open-top area that counts is 46 m², because the 6 m² top is removed. The faces table in the calculator marks each removed face as not counted so you can see what was left out.
Units matter. Surface area is in square units, so a length in centimetres gives square centimetres, and converting one unit of area is not the same as converting a length: 1 m² is 10,000 cm². The calculator lets each length use its own unit and converts everything first. A cylinder of radius 200 cm and height 3 m gives 628,318.5307 m², the same as radius 2 m, because 200 cm is 2 m.
Where Surface Area Is Used
Surface area decides how much material covers an object. It sets the amount of paint for a tank or a wall, the sheet metal for a duct, the fabric for a tent and the insulation for a pipe. It also controls heat and chemistry: a body loses or gains heat through its surface, so a smaller surface for the same volume keeps heat in, which is why spheres are the most efficient shape for storage. When you also need the capacity of the same container, use the volume calculator, and to turn an area into a paint quantity, use the paint calculator.
Frequently asked questions
What is the difference between total and lateral surface area?
Total surface area is every face of the solid, bases included. Lateral surface area is the sides only. For a cylinder it is 2πrh, and for a box it is 2h(l + w).
How do I find the surface area of an open tank?
Add the area of the faces that exist. For a lidless cylinder that is πr² + 2πrh, and for a lidless box it is the floor plus the four walls. Use the calculator's open-faces setting to remove the top or both ends.
Why does a cone need the slant height?
The curved side of a cone slopes, so its area depends on the sloping length ℓ, not the vertical height. Use ℓ = √(r² + h²), which the calculator computes automatically from the radius and the height.
Can I enter the diameter instead of the radius?
Yes. Tick the diameter option and the tool halves it before using the formulas. You can also mix units, for example a radius in centimetres and a height in metres.
What units is surface area measured in?
Square units: m², cm², mm², ft², in², yd², acres or hectares. The calculator shows the result in all eight so you can copy the one you need.
Related tools: the volume calculator gives capacity, the paint calculator turns area into paint, and the math directory lists every math tool on the site.