# What is the name of a solid enclosed by four planes and a spherical surface?

I am in the process of writing a paper I ran into a solid that I have no name for. This solid is the volume enclosed by four planes and a spherical surface. The four planes intersect at the sphere center and the arcs created by the intersection between the sphere and the planes are great circle segments. It looks like a pyramid with a rounded base, but I am wondering if there is a formal name.

The edges of the solid are shown in blue. Vertices are shown as blue circles, not including the sphere center.

The formal name is spherical pyramid, dated back to $$19^{th}$$ century.

Quoting from page 186 of Elements of geometry and trigonometry Legendre, A.M. (Adrian Marie) (an online copy of English translation of the book by Charles Davies can be found here)

A spherical pyramid is a portion of the solid sphere, included between the planes of a solid angle whose vertex is the center. The base of the pyramid is the spherical polygon intercepted by the same planes.

Another quote from Elements of Geometry and Trigonometry: From the Works of A. M. Legendre (1867) (this is from google book search)

A Spherical Pyramid is a portion of a sphere bounded by a spherical polygon and sectors of circles whose common center is the center of the sphere.

The spherical polygon is called the base of the pyramid, and the center of the sphere is called the vertex of the pyramid.

• Thank you for responding so quickly. Spherical pyramid is the name that I will use. Sep 12, 2019 at 14:31
• I have a follow up question. If the sphere was an ellipsoid, should we the analogous shape an ellipsoidal pyramid? Sep 12, 2019 at 20:04
• @DanielGonzalez Yes and No. Yes, there should be an analogous shape. No, I doubt this has been defined before. If you use that, you should define it first. When you define that, you could point out this is similar to the definition of spherical pyramid by Legendre. Sep 13, 2019 at 1:48
• Thank you, I will make sure to do that. Sep 13, 2019 at 17:06

I suppose you might call it a "solid angle", or even a "rectangular solid angle". Properly speaking, in at least one definition, a solid angle is a measurable subset of the unit sphere, but this is just the cone on that object from the sphere-center, and I doubt anyone would strenuously object.

You could also, I guess, call it a 3D pie-slice.

I look forward to seeing whether there is a more formal name for this particular object.

• You mentioned you were curious about the nomenclature. We found a cited name! Check out the answer above. Sep 12, 2019 at 18:36
• I saw it and upvoted immediately. MSE can be so much fun! Sep 12, 2019 at 22:04

In Spherical Trigonometry also (3 planes) for instance.. while deriving Sine/Cosine rules no name is given to this solid being talked about. Or at least one that stuck in usage.. High time for coining a new word.

• @achille hui found a great citation and he posted it on this page. Check it out. Sep 12, 2019 at 18:37
• Yes, it sounds so apt and correct, Wonder why its usage was avoided hitherto. Sep 12, 2019 at 21:09