Triangular pyramid

Triangular pyramid
Rank	3
Space	Spherical
Notation
Bowers style acronym	Tripy
Coxeter diagram	ox3oo&#y
Elements
Faces	1 equilateral triangle, 3 isosceles triangles
Edges	3+3
Vertices	1+3
Vertex figures	1 triangle
	3 isosceles triangles
Measures (edge lengths b [base], ℓ [lacing])
Circumradius	$\frac{l}{2\sqrt{1-\frac{b^2}{3l^2}}}$
Volume	$\frac{\sqrt3b^2}{12}\sqrt{l^2-\frac{b^2}{3}}$
Height	$\sqrt{l^2-\frac{b^2}{3}}$
Central density	1
Related polytopes
Army	Tripy
Regiment	Tripy
Dual	Triangular pyramid
Conjugate	Triangular pyramid
Abstract & topological properties
Flag count	24
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	A2×I, order 6
Convex	Yes
Net count	4
Nature	Tame

The triangular pyramid, or tripy, is a type of tetrahedron with 1 equilateral triangle as a base and 3 isosceles triangles as sides. It is a pyramid based on the equilateral triangle.

It generally has 2 edge sizes, the base and side edges.

In vertex figures

Triangular pyramids occur as the vertex figures in 8 nonprismatic uniform polychora (including the truncated pentachoron, truncated tesseract, truncated icositetrachoron, truncated hecatonicosachoron, truncated great grand hecatonicosachoron, quasitruncated tesseract, quasitruncated small stellated hecatonicosachoron, and quasitruncated great grand stellated hecatonicosachoron) and 3 polyhedral prisms (the tetrahedral prism, dodecahedral prism, and great stellated dodecahedral prism).