Question
What is the least number of planes that can enclose a solid? Name the simplest regular polyhedron and verify Euler’s formula for it.
Solution
At least 4 planes can form to enclose a solid. Tetrahedron is the simple polyhedron. Following figure represents a simplest solid, called tetrahedron.
A tetrahedron has:
4 triangular faces, i.e. F = 4
4 vertices, i.e. V = 4
6 edges, i.e. E = 6
Now, substituting the values of F, V and E in Euler's formula, i.e.
F + V = E + 2
we have
4 + 4 = 6 + 2
8 = 8, which is true.
Thus, Euler's formula is verified for a tetrahedron.
A tetrahedron has:
4 triangular faces, i.e. F = 4
4 vertices, i.e. V = 4
6 edges, i.e. E = 6
Now, substituting the values of F, V and E in Euler's formula, i.e.
F + V = E + 2
we have
4 + 4 = 6 + 2
8 = 8, which is true.Thus, Euler's formula is verified for a tetrahedron.
