# What is the volume of this 3d shape?

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: 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.

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.
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.
• Ooh, that formula for $\tan\frac\Omega2$ is beautiful.