Verify Euler's formula for each of the following polyhedrons:

(i) In the given polyhedron:

Edges $\mathrm{E}=15$

Faces $\mathrm{F}=7$

Vertices $\mathrm{V}=10$

Now, putting these values in Euler's formula:

LHS : F+V

$=7+10$

$=17$

LHS : E+2

$=15+2$

$=17$

LHS = RHS

Hence, the Euler's formula is satisfied.

(ii) In the given polyhedron:

Edges $\mathrm{E}=16$

Faces $\mathrm{F}=9$

Vertices $\mathrm{V}=9$

Now, putting these values in Euler's formula:

RHS : F+V

$=9+9$

$=18$

LHS : E+2

$=16+2$

$=18$

LHS $=$ RHS

Hence, Euler's formula is satisfied.

(iii) In the following polyhedron:

Edges $\mathrm{E}=21$

Faces $\mathrm{F}=9$

Vertices $\mathrm{V}=14$

Now, putting these values in Euler's formula:

$L H S: \mathrm{F}+\mathrm{V}$

$=9+14$

$=23$

$R H S: \mathrm{E}+2$

$=21+2$

$=23$

This is true.

Hence, Euler's formula is satisfied.

(iv) In the following polyhedron:

Edges $\mathrm{E}=8$

Faces $\mathrm{F}=5$

Vertices $\mathrm{V}=5$

Now, putting these values in Euler's formula:

LHS : $\mathrm{F}+\mathrm{V}$

$=5+5$

$=10$

RHS : $\mathrm{E}+2$

$=8+2$

$=10$

LHS $=$ RHS

Hence, Euler's formula is satisfied.

(v) In the following polyhedron:

Edges $\mathrm{E}=16$

Faces $\mathrm{F}=9$

Vertices $\mathrm{V}=9$

Now, putting these values in Euler's formula:

LHS : F+V

$=9+9$

$=18$

RHS : E+2

$=16+2$

$=18$

LHS $=$ RHS

Hence, Euler's formula is satisfied.