The cylinder surface area formula — SA = 2πrh + 2πr² — comes from unrolling the shape into a flat net of three simple pieces.
From 3D Shape to 2D Net
Every cylinder is made of exactly three parts when flattened: two identical circles (the top and bottom bases) and one rectangle (the lateral face that wraps around).
Deriving the Lateral Area
Key insight: The unrolled rectangle's width = circumference of the circle = 2πr
Height = h
Lateral Area = 2πr × h = 2πrh
Height = h
Lateral Area = 2πr × h = 2πrh
Adding the Bases
Each base = πr²
Two bases = 2πr²
Total Cylinder SA = 2πrh + 2πr² = 2πr(h + r)
Example: r = 4 cm, h = 10 cm
SA = 2π(4)(10 + 4) = 2π(56) = 351.86 cm²
Two bases = 2πr²
Total Cylinder SA = 2πrh + 2πr² = 2πr(h + r)
Example: r = 4 cm, h = 10 cm
SA = 2π(4)(10 + 4) = 2π(56) = 351.86 cm²
Open vs Closed Cylinder Variants
| Variant | Formula | Real-World Example |
|---|---|---|
| Closed (both ends) | 2πrh + 2πr² | Sealed can, gas tank |
| Open top | 2πrh + πr² | Cup, bucket, pool |
| Open both ends | 2πrh | Pipe, tube, tunnel |
The Key Insight: When reassembled, the rectangle's width must exactly match the circle's circumference (2πr) — this is why π appears in the lateral area formula.
Why This Derivation Matters
Understanding the net makes it impossible to forget the formula — you can always re-derive it on an exam by picturing the three flat pieces.
Calculate Cylinder Surface Area
Use our Cylinder Calculator for instant results. See cone net derivation, SA vs lateral area, and pipe insulation.