Question

if a Polygon has 10 faces and 16 edges then the no. of vertices it has __________​

Answer 1

${\large{\fcolorbox{aqua}{navy}{\fcolorbox{yellow}{blue}{\bf{\color{yellow}{⫷Question⫸}}}}}}$

If a Polygon has $10$ faces and $16$ edges, then the no. of vertices it has ?

${\large{\fcolorbox{aqua}{navy}{\fcolorbox{yellow}{blue}{\bf{\color{yellow}{⫷Given⫸}}}}}}$

The polygon has $10$ faces and $16$ edges.

${\large{\fcolorbox{aqua}{navy}{\fcolorbox{yellow}{blue}{\bf{\color{yellow}{⫷To\:find⫸}}}}}}$

The no. of vertices.

${\large{\fcolorbox{aqua}{navy}{\fcolorbox{yellow}{blue}{\bf{\color{yellow}{⫷Euler's\: Relation⫸}}}}}}$

$\mathtt{\red{(F-E+V)=2}}$

$\mathtt{\blue{Where,}}$

• $\mathtt{\orange{F=No.\:of\:faces}}$

• $\mathtt{\orange{E=No.\:of\:edges}}$

• $\mathtt{\orange{V=No.\:of\:vertices}}$

${\large{\fcolorbox{aqua}{navy}{\fcolorbox{yellow}{blue}{\bf{\color{yellow}{⫷Solution⫸}}}}}}$

Let the number of vertices be $x.$

According to Euler's Relations,

$\mathtt{\red{(F-E+V)=2}}$

$\mathtt{\red{➳\:(10-16+x)=2}}$

$\mathtt{\red{➳\:(x-6)=2}}$

$\mathtt{\red{➳\:x=(2+8)}}$

$\mathtt{\red{➳\:x=8}}$ $\purple{✯}$

${\large{\fcolorbox{aqua}{navy}{\fcolorbox{yellow}{blue}{\bf{\color{yellow}{⫷Hence⫸}}}}}}$

$\mathtt{x=8}$

So, no. of vertices $\mathtt{=x=8}$

${\large{\fcolorbox{aqua}{navy}{\fcolorbox{yellow}{blue}{\bf{\color{yellow}{⫷Therefore⫸}}}}}}$

The polygon has $8$ vertices.

${\large{\fcolorbox{aqua}{navy}{\fcolorbox{yellow}{blue}{\bf{\color{yellow}{⫷Done⫸}}}}}}$

$\bold\green{\boxed{{\blue{✿}}{\pink{H}}{\blue{O}}{\green{P}}{\red{E}}{\purple{  }}{\orange{T}}{\pink{H}}{\blue{I}}{\green{S}}{\red{  }}{\purple{H}}{\orange{E}}{\pink{L}}{\blue{P}}{\green{S}}{\red{  }}{\purple{Y}}{\orange{O}}{\pink{U}}{\blue{✿}}}}$

$\bold\red{\boxed{{\blue{✿}}{\pink{H}}{\blue{A}}{\green{V}}{\red{E}}{\purple{  }}{\orange{A}}{\pink{  }}{\blue{N}}{\green{I}}{\red{C}}{\purple{E}}{\purple{  }}{\orange{D}}{\pink{A}}{\green{Y}}{\blue{✿}}}}$