Surface Area Calculator: Surface Area of a Cylinder and Closed Solids
Closed cylinder
SA = 2πr² + 2πrh
Radius r · Height h
A closed cylinder has two circular bases and one curved wall.
SA = 2πr² + 2πrh
The fixed axonometric view makes the rear wall translucent; it still contributes to total area. The wall strips illustrate unrolling, and their projected widths change as the curved surface flattens; this is not a rigid stretching proof. Circular boundaries are sampled polygons. Common scale is normalized from the dimensions; extreme proportions may be difficult to see, and the diagram is not to scale.
Worked example you can change
SA = 2πr² + 2πrh
- SA = 2 × π × 2² + 2 × π × 2 × 3
- SA ≈ 62.8318530717959 square units
Calculator
Enter complete decimal numbers using one common unit for dimensions; changing the unit label does not convert numbers. Decimal answers are shown to 15 significant digits.
- Total closed surface area
- 62.8318530717959 cm²
Formula variants
| Shape or method | Formula | Meaning |
|---|---|---|
| Closed cylinder | SA = 2πr² + 2πrh | Include both circular bases and the wall, whose net is a rectangle of width 2πr and height h. |
| Rectangular prism | SA = 2(lw + wh + hl) | Add three matching pairs of rectangular faces. |
| Cube | SA = 6s² | All six faces are squares with area s². |
| Closed right cone | SA = πr² + πrℓ; ℓ = √(r² + h²) | Include the circular base and curved side; perpendicular height and slant height differ. |
| Sphere | SA = 4πr² | Static explanation: four times the area of a great-circle disc; a sphere has no distortion-free flat net. |
Questions
What is the surface area of a cylinder formula?
For a closed cylinder, total surface area is 2πr² + 2πrh, including both bases. Radius 2 and height 3 give 20π square units, approximately 62.8318530717959.
Does this calculate an open container?
No. Cylinder, prism, cube and cone modes include every face or base of the closed solid. Do not use the result for an open container without accounting for omitted faces.
Why does a cone need slant height?
The curved side area is πrℓ. Enter perpendicular height h; the calculator derives ℓ = √(r² + h²) for a right cone.
Can a sphere unfold into a net?
No distortion-free flat net exists for a sphere. Its panel gives a static formula explanation rather than an unfolding proof.
Does the unit selector convert measurements?
It labels a common input unit and squared output unit. Convert measurements yourself before entering them; dimensions are mathematical inputs, not an uncertainty model.
Sources and method
Calculate total closed surface area of a cylinder, rectangular prism, cube, cone or sphere with worked formulas and a cylinder net.
We calculate with 40 significant digits and round only the displayed result. Diagrams are labelled explanations and are not measurement drawings.
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