Question

Can anyone help me to determine the spherical cap volume with only 2 variables available, which are the sphere radius and the sphere cap diameter

Answer 1

What are the variables?

Answer 2

spherical cap diameter and sphere radius

Answer 3

do you have a diagram?

Answer 4

you could do it as a volume integral

if you have diameter D and radius R, you can say that \(\sin \phi = \dfrac{D}{2R}\) where \(\phi\) is the polar angle.

so the volume of the cone, ie less the cap, is:

[\( \star\)] \(\large 2\pi \; \int\limits_{r = 0}^{R} \; \int\limits_{\phi = 0}^{\arcsin (\frac{D}{2R})}\; r^2 \sin\phi \,dr \,\, d\phi .\)

LESS [\(\square\)] the volume of the cone from the usual formula...

i can't get wolfram to do \(\star\) for me but i think it should be:

\(2 \pi \dfrac{R^3}{3} \left(1-\sqrt{\dfrac{1-D^2}{4 R^2}}\right)\)

as it is doing the inner integral as follows:
http://www.wolframalpha.com/input/?i=+int_%7Bp+%3D+0%7D%5E%7Barcsin%28D%2F%282R%29%29+%7D++++sin%28p%29++++dp+


and \(\square\) is \(\dfrac{1}{3}\pi \left(\dfrac{D}{2} \right)^2 \sqrt{R^2 - \left(\dfrac{D}{2}\right)^2}\)

or summat like that :-)

Answer 5

It seems to me the problem is calculating the spherical cap height when just the sphere radius and the spherical cap radius are known.
I have made a graphic which is attached.
We can easily calculate the small green line (spherical cap radius minus radius) but can we determine the spherical cap HEIGHT from this information?

Answer 6

Heck, the cap height can easily be solved from cap radius and sphere radius.
height2^2 = radius^2 - cap radius ^2

Then finding the cap height is just a matter of subtracting height 2 from the radius.

Answer 7

@wolf1728, indeed i can calculate the spherical cap height by subtracting sphere radius to square root of R^2 - r^2 (based on phytagorean theorem

Answer 8

Yes, it occurred to me too and I just came back to post that! LOL

Answer 9

And that seems a LOT easier than calculating a volume integral.

Answer 10

thank youso much, wolf ..... :)

Answer 11

U R welcome!

Answer 12

now i think i can close this

Answer 13

Okay!

Answer 14

nice meeting u

Answer 15

you were asked to calculate the volume of the spherical cap

Answer 16

https://en.wikipedia.org/wiki/Spherical_cap will help if you just want to use a formula....

Answer 17

But the formula they show is for spherical cap volume IF you know the cap height and radius.
We were determining the formula if you only know the spherical cap radius and the sphere radius.

Answer 18

yep, so now plug the height in to get what they want, ie the volume :p