Andrew M. answered • 04/21/21
Whenever I'm finding the surface area of a cylinder, I imagine rolling up a piece of paper. The rectangular face of the paper gets curled to become the long side of the cylinder while the circles formed at each end become the bases.
So then we have the area of each base, which are circles:
A = πr2
Then we have the area of the long side, which is a curled up rectangle. Normally, a rectangle would simply use:
A = L * W
However, since it's all curled up, our length has become the circumference of our base:
L = 2πr
And the width of our paper has become the height of the cylinder:
W = h
Putting all of this together, the area of the side becomes:
A = 2πr * h
For composite shapes, we find the surface area of each piece, then subtract the overlap, since that's no longer on the surface. If they're touching at the bases, then the smaller base needs to be subtracted from the total twice, since not only is that base no longer on the surface, but it's covering up the same amount of space on the other piece.
I'll do an example problem putting all of this together (this way, you can still check your understanding with your own problem):
Two cylinders, touching at the bases.
First cylinder: height = 5, radius = 2
Second cylinder: height = 7, radius = 4
Surface Area of First Cylinder:
Bases: πr2 = π(2)2 = 4π
Side: 2πr * h = 2π(2) * 5 = 20π
Total: Base + Base + Side = 4π + 4π + 20π = 28π
Second Cylinder:
Bases: π(4)2 = 16π
Side = 2π(4) * 7 = 56π
Total: Base + Base + Side = 16π + 16π + 56π = 88π
Overlap:
Smaller Base: 4π
Total Overlap: 4π + 4π = 8π
Composite Figure:
First Cylinder + Second Cylinder - Overlap = 28π + 88π - 8π = 108π