Edge Length and Volume - Dynamic Geometry 2011

Edge Length and Volume of Five-compounds

Small Rhombicuboctahedron and Its Compound

Small Rhombicuboctahedron and Its Compound

	Small Rhombicuboctahedron	Cube	Octahedron
Edge	$a$	$(1 + \sqrt{2}) a$	$(1 + \frac{3}{\sqrt{2}}) a$
Volume	$\frac{1}{3} (12 + 10 \sqrt{2}) a^{3}$	$(7 + 5 \sqrt{2}) a^{3}$	$\frac{1}{6} (45 + 29 \sqrt{2}) a^{3}$

Ratio
- Edge ratio $= 2 : 2 + 2 \sqrt{2} : 3 + 2 \sqrt{2}$
- Volume ratio $= 24 + 20 \sqrt{2} : 42 + 30 \sqrt{2} : 45 + 29 \sqrt{2}$

Truncated Tetrahedron and Its Compound

Great Rhombicuboctahedron and Its Compound

	Great Rhombicuboctahedron	Smaller Cube	Octahedron	Larger Cube
Edge	$a$	$(1 + 2 \sqrt{2}) a$	$(3 + \frac{3}{\sqrt{2}}) a$	$(3 + \sqrt{2}) a$
Volume	$(22 + 14 \sqrt{2}) a^{3}$	$(25 + 22 \sqrt{2}) a^{3}$	$\frac{9}{2} (7 + 5 \sqrt{2}) a^{3}$	$(45 + 29 \sqrt{2}) a^{3}$

Ratio
- Edge ratio $= 2 : 2 + 4 \sqrt{2} : 6 + 3 \sqrt{2} : 6 + 2 \sqrt{2}$
- Volume ratio $= 44 + 28 \sqrt{2} : 50 + 44 \sqrt{2} : 63 + 45 \sqrt{2} : 90 + 58 \sqrt{2}$

Icosahedron and Its Compound

Icosahedron and Its Compound

	Icosahedron	Tetrahedron
Edge	$a$	$\frac{1}{2} (3 \sqrt{2} + \sqrt{10}) a$
Volume	$\frac{5}{12} (3 + \sqrt{5}) a^{3}$	$(3 + \frac{4 \sqrt{5}}{3}) a^{3}$

Ratio
- Edge ratio $= 2 : 3 \sqrt{2} + \sqrt{10}$
- Volume ratio $= 15 + 5 \sqrt{5} : 36 + 16 \sqrt{5}$

Compound of Cube and Octahedron

Icosahedron and Its Compound

	Icosahedron	Octahedron
Edge	$a$	$\frac{1}{4} (3 \sqrt{2} + \sqrt{10}) a$
Volume	$\frac{5}{12} (3 + \sqrt{5}) a^{3}$	$\frac{1}{6} (9 + 4 \sqrt{5}) a^{3}$

Ratio
- Edge ratio $= 4 : 3 \sqrt{2} + \sqrt{10}$
- Volume ratio $= 15 + 5 \sqrt{5} : 18 + 8 \sqrt{5}$

Compound of Five Cubes

Rhombic Triacontahedron and Its Compound

	Rhombic Triacontahedron	Cube
Edge	$a$	$\frac{2}{\sqrt{5}} \sqrt{5 + 2 \sqrt{5}} a$
Volume	$4 \sqrt{5 + 2 \sqrt{5}} a^{3}$	$(\frac{16 + 8 \sqrt{5}}{5}) (\sqrt{5 + 2 \sqrt{5}}) a^{3}$

Ratio
- Edge ratio $= \sqrt{5} : 2 \sqrt{5 + 2 \sqrt{5}}$
- Volume ratio $= 5 : 4 + 2 \sqrt{5}$