# Spherical Sector Volume

I'm trying to find the volume of a spherical sector without knowing the height of the cap. Wikipedia provides this formula:

And says: "where φ is half the cone angle, i.e., the angle between the rim of the cap and the direction to the middle of the cap as seen from the sphere center."

http://en.wikipedia.org/wiki/Spherical_sector

This sentence is kind of ambiguous and I was wondering if someone could just clarify it for me. Should Phi be inputted as the angle between the base of the cap and the side of the sector divided by 2?

• On further inspection, the formula seems wrong regardless of what angle you put into Phi. Does anyone know a way of working out the volume of the spherical sector without knowing the height of the cap. Or indeed finding the height of the cap? Oct 29 '14 at 3:19

$2\phi$ is the angle of the cone (spherical sector)

From Wikipedia: $V=\frac{2\pi r^2 h}{3}$ ...1

From the cone:

$\frac{r-h}{r}=cos\phi$

After simplifying

$h=r(1-cos\phi)$

Replace in 1 to get

$V=\frac{2\pi r^3}{3}(1-cos\phi)$

• I'm pretty confused, I just tried both of these formulas to calculate height and got completely different answers... upload.wikimedia.org/math/a/9/a/… I put the radius as 1 and theta as 90 and got for the first one: 0.47467801118227027, and the second one: 0.14909647546588156 Oct 29 '14 at 3:39
• Can you demonstrate some of your working? Oct 29 '14 at 3:41
• Sure, it's in computer code. Hopefully it's not too difficult to work out if you've never seen code before. It maybe easier to read if you copy and paste into notepad and space it onto seperate lines. l is radius of circle (or length of side of cone). h is height of cone. h = l * Math.sin(theta); r = sqrt(Math.pow(l,2) - (h^2)); cone volume = pi * (r^2) * (h^3); cap height (method one) = l * (1 - cos(theta/2)); cap height (method two) = l - sqrt((l^2) - ((r*2^2)/4)); scv = ((2 * pi * l^2)/3) * cap height; csv = scv - cv; // circular segment volume Oct 29 '14 at 4:13
• So basically, if I use the two formula's for height on this page: en.wikipedia.org/wiki/Circular_segment. And enter theta as 90, R as 1, c as (2*R*Cos(45) I get for the first equation... 0.47467801118227027 and for the second... 0.14909647546588156 Oct 30 '14 at 4:23
• @Varrick What you forgot is that in your software, cos(45) is the cosine of $45$ radians, not $45$ degrees. So you are getting answers that have almost nothing to do with the problem you wanted to solve. If you had set c = 2*R*sin(45) (because the correct formula for the radius of the disk is $2R\sin(\theta/2)$), you would at least have gotten the same answer twice, although it would still not be the answer to the question you meant to ask. Jan 20 '20 at 19:00