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. eulers formula is f + v - e = 2. polyhedron tetrahedron hexahedron (cube) octahedron dodecahedron icosahedron faces (f) 4 6 8 12 20 vertices (v) 4 8 6 20 12 edges (e) 6 12 12 30 30 value of f + v - e 2 2 2 2 2

Explanation:

Step1: Recall Euler's formula

Euler's formula for polyhedra is $F + V - E=2$, where $F$ is the number of faces, $V$ is the number of vertices, and $E$ is the number of edges.

Step2: Check given values for each polyhedron

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 $F+V - E=20+12 - 30=2$.

Conclusion:

For all five regular polyhedra (tetrahedron, hexahedron, octahedron, dodecahedron, icosahedron), the value of $F + V - E$ is equal to 2, which verifies Euler's formula for these polyhedral shapes.

Answer:

The value of $F + V - E$ is 2 for all the five regular polyhedra as shown by the calculations above, confirming Euler's formula.