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The surface area of a three-dimensional shape is the total area of all of the faces of the
shape. There are some rules for the surface area of specific shapes, but I actually think
it’s better and easier just to work it out carefully each time. Otherwise, like for perimeter,
there’s too many rules and it’s too easy to get confused about which one you want.
Just figure out what the net of the shape would look like, what the individual flat
faces would be, and then calculate each one separately before adding them up for the total
surface area.
There are some shapes, though, that have curved faces that you can’t easily work out. And
they do have fancy rules that you can learn, so you don’t have to do calculus every time
you want to use one. But it’s really important that you understand where these formulas come
from and what the different parts of them mean, so that you can apply them appropriately
when you’re solving problems.
Take this cylinder for example. Although it does have a curved face, when you flatten
it out that face is actually just a rectangle. The width of the rectangle is just the height
of the cylinder. And the length of the rectangle is the same as the circumference of the circle
at the base. See that? The area of that rectangle is the height of the cylinder multiplied by
the circumference of the circle. And then the total surface area is just that plus the
two circles, one on the top and one on the bottom. So now you understand this formula,
you can use it in a situation where, for example, you don’t need the top circle, like if you’re
painting an open can. Or where you only need the curved surface. Or whatever.
Likewise, look at this cone. I won’t try to explain why this bit of the formula works
for the curved surface, which comes out as a sector of a circle when you flatten it out.
(It has to do with using radians to measure angles instead of degrees.) You just need
to remember to use the slant height for the surface area, not the perpendicular height.
But this first bit is just the area of the circle at the bottom. So now you know which
part to omit if you don’t need the base, for example if you’re calculating the area
of material for a conical tent or something like that.
And then there are the spheres. The surface area of a whole sphere is four pi r squared.
You need to learn this one, along with the rule for the cone that I just showed you.
But the rest you can work out. An open hemisphere is simply half of a sphere, so it’s two
pi r squared. And then a closed hemisphere, with another flat circular face across the
middle where the sphere was cut off, makes another pi r squared, so the total is three
pi r squared.