Chapter 12.4, Problem 1BE

a.

To determine

To verify : Euler’s formula for the given figure.

Answer to Problem 1BE

The Euler’s form is verified.

Explanation of Solution

Given information :

For any solid figure with polygons for faces, Euler’s formula, $F+V-E=2$ , must hold, where F , V, and E stands for the number o faces, vertices, and edges respectively.

Calculation :

Since, the given figure is a cube.

In cube total numbers of faces are 6. Therefore,

  $F=6$

Total numbers of vertices are 8. Therefore,

  $V=8$ and

Total numbers of edges are 12. Therefore,

  $E=12$

Substitute these values in Euler’s equation and check:

  $\begin{array}{l}F+V-E=6+8-12\\ F+V-E=14-12\\ F+V-E=2\end{array}$

Hence,

The Euler’s form is verified.

b.

To determine

To verify : Euler’s formula for the given figure.

Answer to Problem 1BE

The Euler’s form is verified.

Explanation of Solution

Given information :

For any solid figure with polygons for faces, Euler’s formula, $F+V-E=2$ , must hold, where F , V, and E stands for the number o faces, vertices, and edges respectively.

Calculation :

Since, the given figure is a octahedron.

In octahedron total numbers of faces are 8. Therefore,

  $F=8$

Total numbers of vertices are 6. Therefore,

  $V=6$ and

Total numbers of edges are 12. Therefore,

  $E=12$

Substitute these values in Euler’s equation and check:

  $\begin{array}{l}F+V-E=8+6-12\\ F+V-E=14-12\\ F+V-E=2\end{array}$

Hence,

The Euler’s form is verified.

c.

To determine

To verify : Euler’s formula for the given figure.

Answer to Problem 1BE

The Euler’s form is verified.

Explanation of Solution

Given information :

For any solid figure with polygons for faces, Euler’s formula, $F+V-E=2$ , must hold, where F , V, and E stands for the number o faces, vertices, and edges respectively.

Calculation :

Since, the given figure is a Icosahedron.

In Icosahedron total numbers of faces are 20. Therefore,

  $F=20$

Total numbers of vertices are 12. Therefore,

  $V=12$ and

Total numbers of edges are 30. Therefore,

  $E=30$

Substitute these values in Euler’s equation and check:

  $\begin{array}{l}F+V-E=20+12-30\\ F+V-E=32-30\\ F+V-E=2\end{array}$

Hence,

The Euler’s form is verified.