Chapter 1.7, Problem 34PPS

To determine

To Show: Whether Euler’s formula is true for the given figure.

Answer to Problem 34PPS

The figures expect the figure 6 th do not satisfy the Euler’s formula is not the polyhedron.

Explanation of Solution

Given:

The number of faces $F$ , vertices $V$ , and edges $E$ of a polyhedron are related by Euler's (OY luhrz) Formula: $F+V=E+2$ .

Calculation:

Consider the relation by the Euler’s formula,

  $V-E+F=2$

Since, the first figure is rectangular prism so there are 8 vertices, 12 edges and the 6 faces that is $V=8$ , $E=12$ and $F=6$ .

Then, replace $V,E$ and $F$ by 8, 12 and 6 respectively in the left side of the Euler’s formula.

  $\begin{array}{c}V-E+F=8-12+6\\ =2\end{array}$

This proves the Euler’s formula.

Since, the second figure is square prism so there are 8 vertices, 12 edges and the 6 faces that is $V=8$ , $E=12$ and $F=6$ .

Then, replace $V,E$ and $F$ by 8, 12 and 6 respectively in the left side of the Euler’s formula.

  $\begin{array}{c}V-E+F=8-12+6\\ =2\end{array}$

Since, the third figure is triangular prism so there are 6 vertices, 9 edges and the 5 faces that is $V=8$ , $E=12$ and $F=6$ .

Then, replace $V,E$ and $F$ by 8, 12 and 6 respectively in the left side of the Euler’s formula.

  $\begin{array}{c}V-E+F=6-9+5\\ =2\end{array}$

Since, the sixth figure is pentagonal pyramid so there are 5 vertices, 8 edges and the 5 faces that is $V=5$ , $E=8$ and $F=5$ .

Then, replace $V,E$ and $F$ by 5, 10 and 5 respectively in the left side of the Euler’s formula.

  $\begin{array}{c}V-E+F=5-8+5\\ =2\end{array}$

The sixth figure is the cylinder as that is not the polyhedron.

Thus, the figures expect the figure 6 th do not satisfy the Euler’s formula is not the polyhedron.