Question

find the solution using euler's formula (2) can a polyhedron have 20 faces,30edges and 12 vertices ?prove by eulera formul​

Answer 1

$\huge\boxed{\textsf{\textbf{\pink{Given\::-}}}}$

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• A polyhedron have 20 faces, 30 edges and 12 vertices.

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$\huge\boxed{\textsf{\textbf{\green{To\:Prove\::-}}}}$

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• Prove the statement i.e, polyhedron have 20 faces, 30 edges and 12 vertices with euler's formula.

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$\huge\boxed{\textsf{\textbf{\blue{Proof\::-}}}}$

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$\large\underbrace{\underline{\sf{\bigstar\:Understanding\:the\:Question\::-}}}$

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• Here, the question gives faces of polyhedron, edges of polyhedron, and vertices of polyhedron. We have to prove the given statement with euler's formula.

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• We clearly know that if we are given faces, edges and vertices of polyhedron, then we use well known formula, i.e, euler's formula :-

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• $\huge\underline{\boxed{\bf{\red{V - E + F = 2}}}}$

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$\small \frak {where} \begin{cases} & \sf {V\:denotes\:\bf{Vertices}\:\sf{of\:polyhedron}} \\ & \sf{E\:denotes\:\bf{Edges}\:\sf{of\:polyhedron}} \\ & \sf {F\:denotes\:\bf{Faces}\:\sf{of\:polyhedron}}\end{cases}$

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$\small \frak {we\:have} \begin{cases} & \sf {Vertices\:\big(V\big) = \bf{12}} \\ & \sf{Edges\:\big(E\big) = \bf{30}} \\ & \sf {Faces\:\big(F\big) = \bf{20}}\end{cases}$

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$\underline{\sf{\bigstar\:Putting\:all\:known\:values\:in\:formula\::-}}$

$\\ :\implies \:\sf 12 - 30 + 20 = 2$

$\\ :\implies \:\sf -18 + 20 = 2$

$\\ :\implies \:\boxed{\bf{\purple{ 2 = 2}}}\:\bigstar$

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$\huge\boxed{\textsf{\textbf{\orange{Hence,\:Proved!}}}}$

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∴ Hence, given statement is correct!

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Answer 2

$\sf\underline{\red{\:\:\:\: Given:-\:\:\:\:}}$

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The polyhedron has 20 faces, 30 edges and 12 vertices.

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$\sf\underline{\red{\:\:\: Need\:To\: Prove:-\:\:\:}}$

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We need to prove this statement by using Euler's formula.

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$\sf\underline{\red{\:\:\:Proof:-\:\:\:}}$

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Here in this question we are given with the faces, edges of polyhedron as well as vertices of polygon respectively. That is,

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$\frak{Here}\begin{cases}\sf{\:\;\; F = 20}\\\\\sf{\;\;\; V = 12}\\\\\sf{\;\;\; E = \: 30}\end{cases}$

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We know that if we are given with the faces, edge & vertices of polyhedron, we have the required formula, that is,

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$\sf{:\implies V - E + F = 2}$⠀

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⠀⠀⠀⠀ Here V is the vertices of polyhedron, E is the number edge and F is the Faces of polyhedron, And here in this question we have V = 12, E = 30 and F = 20. So by using the Euler's polyhedron formula we can easily proof.

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By using the formula and substituting all the given values in the formula, we get:

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$\sf{:\implies 12 - 30 + 20 = 2}$

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$\sf{:\implies -18 + 20 = 2}$

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$\sf{:\implies \blue{\boxed{\pmb{\bf{ \pink{2 = 2}}}}}}$

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Therefore, this statement is true.