Edge Length and Volume - Dynamic Geometry 2011

Octahedron and Its Compound

Truncated Tetrahedron and Its Compound

Ratio
- Edge ratio $= 1 : 3 : 5$
- Volume ratio $= 1 : 9 : 125 = 1 : 3^{2} : 5^{3}$

Cuboctahedron and Its Compound

Truncated Cube and Its Compound

Ratio
- Edge ratio $= 2 : 2 + \sqrt{2} : 4 + 3 \sqrt{2}$
- Volume ratio $= 42 + 14 \sqrt{2} : 42 + 30 \sqrt{2} : 99 + 70 \sqrt{2}$

Truncated Octahedronand and Its Compound

Ratio
- Edge ratio $= 1 : 3 : 2 \sqrt{2}$
- Volume ratio $= 8 : 9 : 16 = 2^{3} : 3^{2} : 2^{4}$

Icosidodecahedron and Its Compound

Ratio
- Edge ratio $= 3 + \sqrt{5} : 2 \sqrt{6 + 3 \sqrt{5}} : 6 + 2 \sqrt{5}$
- Volume ratio $= 220 \sqrt{3 + \sqrt{5}} + 96 \sqrt{15 + 5 \sqrt{5}} : 24 \sqrt{2} (15 + 7 \sqrt{5}) : 280 \sqrt{3 + \sqrt{5}} + 120 \sqrt{15 + 5 \sqrt{5}}$

Truncated Dodecahedron and Its Compound

	Truncated Dodecahedron	Dodecahedron	Icosahedron
Edge	$a$	$\frac{2 + \sqrt{9 - 3 \sqrt{5}}}{2} a$	$\frac{1 + 3 \sqrt{5}}{2} a$
Volume	$\frac{5}{12} (99 + 47 \sqrt{5}) a^{3}$	$\frac{1}{8} (75 + 75 \sqrt{\frac{3}{2}} + 4 \sqrt{\frac{27745}{32} + \frac{5535 \sqrt{\frac{3}{2}}}{8}}) a^{3}$	$\frac{1}{12} (705 + 355 \sqrt{5}) a^{3}$

Ratio
- Edge ratio $= 2 : 2 + \sqrt{9 - 3 \sqrt{5}} : 1 + 3 \sqrt{5}$
- Volume ratio $= 990 + 470 \sqrt{5} : 225 + 225 \sqrt{\frac{3}{2}} + 6 \sqrt{\frac{27745}{8} + \frac{5535 \sqrt{\frac{3}{2}}}{2}} : 1410 + 710 \sqrt{5}$

Truncated Icosahedron and Its Compound

Ratio
- Edge ratio $= 2 : 6 : 5 \sqrt{5} - 7$
- Volume ratio $= 125 + 43 \sqrt{5} : 135 + 45 \sqrt{5} : 385 - 47 \sqrt{5}$

Dynamic Geometry