Question

Using Euler's formula, Find the value of $w,x,y$and $z$.

Faces	$w$	$7$	$25$	$9$
Vertices	$8$	$x$	$9$	$z$
Edges	$10$	$9$	$y$	$11$

Answer 1

To find the value of $w,x,y$and $z$, apply Euler's formula: $V+F-E=2$ in each column.

In first column,

Faces = $w$

Vertices = $8$

Edges = $10$

apply Euler's formula:

$$\begin{gathered} V+F-E=2 \\ 8+w-10=2 \\ w-2=2 \\ w=2+2 \\ w=4 \end{gathered}$$

In second column,

Faces = $7$

Vertices = $x$

Edges = $9$

apply Euler's formula:

$$\begin{gathered} V+F-E=2 \\ x+7-9=2 \\ x-2=2 \\ x=2+2 \\ x=4 \end{gathered}$$

In third column,

Faces = $25$

Vertices = $9$

Edges = $y$

apply Euler's formula:

$$\begin{gathered} V+F-E=2 \\ 9+25-y=2 \\ 34-y=2 \\ y=34-2 \\ y=32 \end{gathered}$$

In fourth column,

Faces = $9$

Vertices = $z$

Edges = $11$

apply Euler's formula:

$$\begin{gathered} V+F-E=2 \\ z+9-11=2 \\ z-2=2 \\ z=2+2 \\ z=4 \end{gathered}$$