# Finding surface area of cone in spherical coordinates

1. Jan 26, 2012

### ninevolt

Hello everyone,

I recently tried to find the surface area of a hollow cone (there is no base, like an ice cream cone) using spherical coordinates. With cylindrical coordinates I was able to do this easily using the following integral:

$\int \int \frac{R}{h}z \sqrt{\frac{R^{2}}{h^{2}} + 1} dz d\theta$
Where:
R = radius of the base
h = height of the cone
(R/h)z = radius of cone at specific z

$\sqrt{\frac{R^{2}}{h^{2}} + 1}$ - the ds element across the slanted side of the cone

and I will obtain the correct answer for the surface area of a cone:
$\pi R \sqrt{h^{2} + R^{2}}$

but when I try to do the same integral in spherical coordinates I obtain different results
I use the following integral:
$\int \int \rho^{2} sin(\theta) d\rho d\phi$

What am I doing wrong?

2. Jan 27, 2012

### tiny-tim

hello ninevolt!

i think you're confusing θ with the (fixed) semi-angle of the cone

(btw, you might also like to try doing it without integration, by slicing the cone and flattening it!)

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