64th Johnson solid (10 faces)
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]
J
64
{\displaystyle J_{64}}
[ 2]
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
a
{\displaystyle a}
A
{\displaystyle A}
V
{\displaystyle V}
[ 1]
A
=
7
3
4
a
2
+
25
+
10
5
4
a
2
≈
4.751
a
2
V
=
15
+
7
5
24
a
3
+
2
12
a
3
≈
1.395
a
3
.
{\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}}}
References
^ a b Berman, Martin (1971), "Regular-faced convex polyhedra", Journal of the Franklin Institute , 291 (5): 329– 352, doi :10.1016/0016-0032(71)90071-8 , MR 0290245 ^ Francis, Darryl (August 2013), "Johnson solids & their acronyms" , Word Ways , 46 (3): 177
External links
