Surface area of a calotte-shape

coljohnmatrix · #1 $Old$ June 29th, 2009, 02:04 PM

Hi there,

Suffering from bit of a mental block here

Hoping someone can help!

I need to figure out how a step in a derivation from a textbook is done...basically, how exactly do you get to the point where you define the yellow area in the calotte-shaped area shown below as being equal to 2*pi*r *sin phi r d phi ? I presume that the 2*pi*r term arises from the circumference of the circle? The rest, I'm not so sure of...

Like I said, I've got a total mental block - I guess it's probably relatively straightforward!

Many thanks in advance!

HallsofIvy · #2 $Old$ June 29th, 2009, 04:04 PM

For "small $d\phi$ ", you can ignore the "curvature" of the sphere: that is a cylinder. Its area is the area of the rectangle you get by "unfolding" it. It is a rectangle with height $d\phi$ and width equal to the circumference of the circle, $2\pi$ times the radius $= 2\pi r sin \phi$ .

coljohnmatrix · #3 $Old$ June 30th, 2009, 02:49 AM

Thanks! Mental block removed! Much appreciated