Question

Quick! Without looking it up. What is the formula for the surface area of a cone?

Answer 1

Am I allowed to use calculus?

Answer 2

Sure

Answer 3

Get a line y=mx and rotate it around the x-axis from 0 to h.

\[\int_0^h 2\pi m xdx = \pi mh^2\]
Then cap it off with a circle, which has radius \(r=mh\) so

\[A = \pi m h^2 +\pi (mh)^2\]

I dunno maybe I messed up if that's not right I'm really avoiding the way I wanna do it which is in spherical coordinates. If you want in terms of the angle you can substitute \(m=\tan \theta\). Should work I think.

Answer 4

Curious: Why do you say "without looking it up?" There definitely is a standard formula for the surface area of a cone (lateral area only). What's your motive here? Want to know how to derive the formula, or want to avoid having to look it up, or...what?

Answer 5

Please explain where you're coming from.

Answer 6

Or alternatively we can replace:
\[m = \tan \theta = \frac{r}{h}\]which is the radius and height to get:
\[A = \pi rh + \pi r^2 =\pi r(r+h)\]
Seems like this might work as long as I did it right, I'm not gonna look it up to check either lol.

Answer 7

I googled "surface area of a solid of revolution" and got lots of nice, juicy results regarding finding the surface area formula using calculus.

Answer 8

Yeah but he said without looking it up lol

Answer 9

I was doing one of those optimization problems that involved cones, and one of it required the surface area of a cone. I tried to figure out the surface area algebraically, but couldn't and didn't realize it was this complicated. I just wanted to know a simple way, but apparently involves integration, which I haven't learned yet. So yeah, thanks for attempting the challenge.

Answer 10

I guess I will just memorize the formula for now...

Answer 11

@mathmale @kainui

Answer 12

You can try decomposing a cone into simpler geometric regions. A cone can be broken up into its circular base while its lateral "face" is a circular sector.

So given a right cone with height \(h\), base radius \(r\), and slant height \(s=\sqrt{h^2+r^2}\). This slant height determines the radius of the sector that makes the lateral face.

The area of the base is easy enough: \(\pi r^2\).

The area of the lateral face is the area of the circular sector, which is \(\dfrac{s\ell}{2}=\pi r\sqrt{h^2+r^2}\).

Putting everything together, the surface area of a right cone is then \(\pi r^2+\pi r\sqrt{h^2+r^2}\).