Question

many crystals and biological structures are constructed in the shapes of regular polyhedra. leonhard euler investigated a remarkable relationship among the numbers of faces (f), vertices (v), and edges (e) for the five regular polyhedra. complete the table below and then draw a conclusion.
complete the table.

polyhedron	tetrahedron	hexahedron (cube)	octahedron	dodecahedron	icosahedron
vertices (v)
edges (e)
value of f + v - e

Explanation:

Step1: Recall Euler's formula for polyhedra

$F + V - E=2$, where $F$ is the number of faces, $V$ is the number of vertices, and $E$ is the number of edges.

Step2: Identify properties of regular polyhedra

• Tetrahedron:
• $F = 4$, $V=4$, $E = 6$. Then $F+V - E=4 + 4-6=2$.
• Hexahedron (cube):
• $F = 6$, $V = 8$, $E=12$. Then $F + V-E=6 + 8-12=2$.
• Octahedron:
• $F=8$, $V = 6$, $E = 12$. Then $F+V - E=8 + 6-12=2$.
• Dodecahedron:
• $F = 12$, $V=20$, $E = 30$. Then $F+V - E=12+20 - 30=2$.
• Icosahedron:
• $F = 20$, $V=12$, $E = 30$. Then $F+V - E=20 + 12-30=2$.

Polyhedron	Faces (F)	Vertices (V)	Edges (E)	Value of $F + V - E$
Hexahedron (cube)	6	8	12	2
Octahedron	8	6	12	2
Dodecahedron	12	20	30	2
Icosahedron	20	12	30	2

Answer:

Polyhedron	Faces (F)	Vertices (V)	Edges (E)	Value of $F + V - E$
Hexahedron (cube)	6	8	12	2
Octahedron	8	6	12	2
Dodecahedron	12	20	30	2
Icosahedron	20	12	30	2