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I'm wondering if there is an equation that represents the volume of an arbitrary 3d primitive matching this description:

1.) Point at center of sphere 2.) Each edge is the length of the radius 3.) 3 flat sides, 1 arc side

Image:

enter image description here

So it's kind of a sector of a sphere, but instead of a conical shape it's more of a tetrahedral shape, but with a curved end.

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2 Answers 2

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The volume of your figure is $\frac13rA$, where $r$ is the radius of the sphere, and $A$ is the area of the curved end, which is a spherical triangle. A simple formula for $A$ is $r^2(\alpha+\beta+\gamma-\pi)$, where $\alpha$, $\beta$, and $\gamma$ are the angles of the spherical triangle in radians, which are the same as the dihedral angles between the flat faces of your figure. Beyond that, the formula you get will depend on how the three faces or three edges of your figure are specified.

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  • $\begingroup$ In this answer I assumed that by "3 flat sides" you meant that the three faces of the figure that contain the origin are planar, and the remaining curved face is a part of the sphere. $\endgroup$
    – user856
    Commented Jul 13, 2012 at 5:13
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The volume is $\frac13\Omega r^3$, where $\Omega$ is the solid angle subtended at the centre. Wikipedia has several formulas for the solid angle, including the one in Rahul's answer.

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  • $\begingroup$ Ooh, that formula for $\tan\frac\Omega2$ is beautiful. $\endgroup$
    – user856
    Commented Jul 13, 2012 at 5:17

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