Thread: volume of a spherical cap

1. volume of a spherical cap

i need help finding the volume of a spherical cap. the only information i know is that the rim of the cap is 5m and its height is 2m.

I have tried integrating using the formula for a sphere but the problem is I don't know the radius of the sphere from which the cap came and I can't figure out where to go from there. please help. thanks

2. Originally Posted by justonemoremathguy
i need help finding the volume of a spherical cap. the only information i know is that the rim of the cap is 5m and its height is 2m.

I have tried integrating using the formula for a sphere but the problem is I don't know the radius of the sphere from which the cap came and I can't figure out where to go from there. please help. thanks
You don't need the radius; you can determine it from the information given. Using pythagorean theorem:

$\displaystyle r^2=(\frac{5}{2\pi})^2+(r-2)^2$

$\displaystyle r=\frac{25}{16\pi^2}+1$

For the volume, I suggest cylindrical coordinates:

$\displaystyle V=4\int_0^{\frac{\pi}{2}}\int_{\frac{25}{16\pi^2}-1}^{\frac{25}{16\pi^2}+1}\int_0^{\frac{5}{2\pi}}r\ ;dr\;dz\;d\theta$

$\displaystyle V=4\int_0^{\frac{\pi}{2}}\int_{\frac{25}{16\pi^2}-1}^{\frac{25}{16\pi^2}+1}[\frac{r^2}{2}]_0^{\frac{5}{2\pi}}dz\;d\theta$

$\displaystyle V=4\int_0^{\frac{\pi}{2}}\int_{\frac{25}{16\pi^2}-1}^{\frac{25}{16\pi^2}+1}[\frac{25}{8\pi^2}]dz\;d\theta$

$\displaystyle V=4\int_0^{\frac{\pi}{2}}[\frac{25}{4\pi^2}]d\theta$

$\displaystyle V=\frac{25}{2\pi}$