Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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


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.

share|cite|improve this question
up vote 5 down vote accepted

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.

share|cite|improve this answer
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. – Rahul Jul 13 '12 at 5:13

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.

share|cite|improve this answer
Ooh, that formula for $\tan\frac\Omega2$ is beautiful. – Rahul Jul 13 '12 at 5:17

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.