Augmented tridiminished icosahedron

In geometry, the augmented tridiminished icosahedron is a composite convex polyhedron with regular faces, a Johnson solid. It can be obtained by joining a regular tetrahedron to another Johnson solid, the tridiminished icosahedron. The resulting polyhedron has seven equilateral triangular and three regular pentagonal faces.^[1] Out of 92 solids, the augmented tridiminished icosahedron is listed as 64th Johnson solid ${\displaystyle J_{64}}$.^[2]

Augmented tridiminished icosahedron
Type	Johnson J63 – J64 – J65
Faces	1+2×3 triangles 3 pentagons
Edges	18
Vertices	10
Vertex configuration	1(33) 3(3.52) 3(33.5) 3(32.52)
Symmetry group	C3v
Dual polyhedron	-
Properties	convex
Net

The surface area of an augmented tridiminished icosahedron is the sum of the area of seven equilateral triangles and three regular pentagons. Its volume can be obtained by summing the volume of a tridiminished icosahedron and a regular tetrahedron. Let ${\displaystyle a}$ be its edge length. Then its surface area ${\displaystyle A}$ and volume ${\displaystyle V}$ are:^[1] $${\displaystyle {\begin{aligned}A&={\frac {7{\sqrt {3}}}{4}}a^{2}+{\frac {\sqrt {25+10{\sqrt {5}}}}{4}}a^{2}\approx 4.751a^{2}\\V&={\frac {15+7{\sqrt {5}}}{24}}a^{3}+{\frac {\sqrt {2}}{12}}a^{3}\approx 1.395a^{3}.\end{aligned}}}$$