Surface Area of Cylinders – SA = 2πr² + 2πrh
A cylinder's surface area combines its 2 circular ends (each πr²) with its curved lateral surface. Imagine peeling the curved surface off and rolling it flat – it unrolls into a rectangle with height h and width equal to the circle's circumference, 2πr. That rectangle's area is 2πrh, giving the full formula SA = 2πr² + 2πrh.
This “peel and unroll” trick is one of the most elegant ideas in solid geometry – it turns a curved 3D surface into a flat, familiar rectangle, which is exactly why the cylinder's formula includes the circle's circumference formula hidden inside it.
Calculating a Cylinder's Surface Area
SA = 2πr² + 2πrh, where the first term is the two circular ends and the second term is the curved lateral surface.
SA = 2π(3)² + 2π(3)(10) = 18π + 60π = 78π ≈ 245.04 cm².
Real-Life Application
- Can labels: the paper label on a can matches its curved lateral surface area.
- Pipe insulation: calculating material needed to wrap a cylindrical pipe.
- Paint cans: estimating metal needed to manufacture cylindrical containers.
Key Takeaways
- A cylinder's surface area formula is SA = 2πr² + 2πrh.
- The curved surface “unrolls” into a rectangle of width 2πr and height h.
- This unrolling trick connects a cylinder's surface area directly to a circle's circumference.
Practice: Surface Area of Cylinders
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