# Thread: Sphere cap Height calculations

1. ## Sphere cap Height calculations

Hi,

I know the formula for finding the volume of a sphere cap if radius and height are given,But how do I find the height of the cap/dome if only radius and volume are given? ie....r=5.4 volume = 160m3 (air space under dome).

Thanks....Mark

2. Originally Posted by manich44
Hi,

I know the formula for finding the volume of a sphere cap if radius and height are given,But how do I find the height of the cap/dome if only radius and volume are given? ie....r=5.4 volume = 160m3 (air space under dome).

Thanks....Mark
1. Volume of a sphere cap:

$V= \pi r h^2 - \dfrac \pi3 h^3$

EDIT: (r is the radius of the base of the sphere!) Sorry for the confusion!

2. Plug in the values you know. You get a cubic equation which could be solved by using Cardano's formula. You get an approximate result much easier and faster if you use an iterative method, for instance Newton's method.

3. I've got $h \approx 3.46355$

3. ## Re sphere cap

Hi,

I used... pi/6(63.48 + 7.84)2.8 = V 104.6m3 (104.5605811) for the answer im looking for.

How then do I plug the new values known ie r 5.4m and vol 160m3 into that to get the new height,can you show me an example?

Thanks.....Mark

4. Originally Posted by manich44
Hi,

I used... pi/6(63.48 + 7.84)2.8 = V 104.6m3 (104.5605811) for the answer im looking for.

How then do I plug the new values known ie r 5.4m and vol 160m3 into that to get the new height,can you show me an example?

Thanks.....Mark
If I understand you correctly you used:

$V=\frac16 \pi h(3r^2+h^2)=\frac12 \pi r^2 \cdot h + \frac16 \pi \cdot h^3$

Now plug in the known values:

$160 = \frac12 \pi (5.4)^2 \cdot h + \frac16 \pi \cdot h^3$

This is a cubic equation. I already told you how to solve it.

I've got $h \approx 4.62132...$

5. ## Re sphere cap

Hi,

Can someone have a look attachment and let me know where im going wrong,cant get to the height "h"

Thanks.....Mark