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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."

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?

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1 Answer

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$2\phi$ 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

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

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