The formula for the area of the curved part of a frustum is
disproportionately hard to
derive. But, here in this humble node, I shall attempt to
demystify the process of the formula's derivation, for those two or three of you out there who actually care.
Those of you who recently had geometry will remember that the surface area of the curved part of a cone is πrl, where r is the radius of the base of the cone, and l is the slant height of the cone, or the length from the vertex of the cone to a point on the edge of the base. So, to find the surface area of the frustum, all you have to do is subtract the areas of the two cones that make up the top and bottom edges of the frustum, right? So, let's suppose the little and big radii of the frustum are r1 and r2, and the slant height of the frustum is l, and the slant height of the cone that you're subtracting is l1. Then, the surface area of the frustum would be
S = πr2(l1 + l) - πr1l1
S = π(r2l1 + r2l - r1l1)
Now, since the two cones that we're working with are similar, their respective slant heights and radii are proportionate; in other words,
(l1 + l)/r2 = l1/r1
From here, we isolate and get rid of l1.
r1(l1 + l)/r2 = l1
r1l1/r2 + r1l/r2 = l1
r1l/r2 = l1 - r1l1/r2
r1l/r2 = l1(1 - r1/r2)
l1 = (r1l/r2)/(1 - r1/r2) = l{(r1/r2)/[(r2 - r1)/r2]}
l1 = l[r1/(r2 - r1)]
Substituting this into our surface equation, we get
S = π({(r2 - r1)l[r1/(r2 - r1)]} + r2l)
S = π(r1l + r2l)
S = πl(r1 + r2)
And there it is: the formula for the curved surface area of a frustum. Note that the slant height of the subtracted cone is not in the final equation; this is a good thing. For calculus purposes, if you only have one radius to work with (i.e. the frustum is infinitessimally thin) the two radii can be combined as an average (the radius at the center of the frustum):
r = (r1 + r2)/2
S = 2πlr
Now, should you need to know the surface area of the entire frustum, you can just tack on the areas of the top and bottom circles:
S = πl(r1 + r2) + πr12 + πr22
Mommy, my fingers hurt from typing all these angle brackets!
Second in a series of MathRants by PMDBoi.
Node math, people!
NB: this is much easier to read in the printable version. Serifs are good.