How To Find The Volume Of A Sector

Question

I'thousand trying to find the book of a spherical sector without knowing the pinnacle of the cap. Wikipedia provides this formula:

Formula

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

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

This judgement is kind of ambiguous and I was wondering if someone could just clarify information technology for me. Should Phi exist 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 bending you put into Phi. Does anyone know a style of working out the volume of the spherical sector without knowing the elevation of the cap. Or indeed finding the pinnacle of the cap? — October 29, 2014 at 3:xix

Accepted Answer · 2014-10-29 03:28:16Z

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

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

From the cone:

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

After simplifying

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

Supplant in one to become

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

Cone

answered Oct 29, 2014 at three:28

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5

$\begingroup$ I'chiliad 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 every bit 90 and got for the first one: 0.47467801118227027, and the second one: 0.14909647546588156 $\endgroup$

October 29, 2014 at 3:39
$\begingroup$ Can you lot demonstrate some of your working? $\endgroup$

October 29, 2014 at three:41
$\begingroup$ Sure, it'south in computer code. Hopefully it's not too hard to piece of work out if you've never seen code before. It perhaps easier to read if you copy and paste into notepad and infinite it onto seperate lines. l is radius of circle (or length of side of cone). h is height of cone. h = fifty * Math.sin(theta); r = sqrt(Math.pow(l,ii) - (h^2)); cone book = pi * (r^two) * (h^3); cap pinnacle (method one) = l * (1 - cos(theta/ii)); cap height (method two) = l - sqrt((l^2) - ((r*ii^2)/4)); scv = ((2 * pi * l^2)/three) * cap superlative; csv = scv - cv; // round segment volume $\endgroup$

Oct 29, 2014 at four:thirteen
$\begingroup$ So basically, if I use the two formula's for height on this folio: en.wikipedia.org/wiki/Circular_segment. And enter theta as ninety, R every bit 1, c as (2*R*Cos(45) I go for the first equation... 0.47467801118227027 and for the 2d... 0.14909647546588156 $\endgroup$

October xxx, 2014 at 4:23
$\begingroup$ @Varrick What y'all forgot is that in your software, cos(45) is the cosine of $45$ radians, not $45$ degrees. Then you are getting answers that have about nothing to do with the problem y'all wanted to solve. If you had set up c = 2*R*sin(45) (because the correct formula for the radius of the disk is $2R\sin(\theta/2)$), you lot would at least have gotten the aforementioned answer twice, although information technology would yet not exist the answer to the question you lot meant to ask. $\endgroup$

January 20, 2020 at 19:00