Question

1. A polyhedron has 10 edges and 6 vertices . how many faces does this polyhedron have ?​

2. A polyhedron has 6 faces and 8 vertices .how many edges does this polyhedron have .

3. A polyhedron has 40 faces and 60 edges . how many vertices does it have ?

4. How many vertices , faces and edges does a hexagon prism have ?

5. How many vertices , faces and edges does a octagonal pyramid have ?

Answer 1

answer If F=faces,V=vertices,E=edges,

F+V−E=2

i.e, 5+6−E=2,

E=9.

please mark me brainlist answer and give lot of thanks

Answer 2

${\large{\bold{\sf{\underline{1. QUESTION:-}}}}}$

A polyhedron has 10 edges and 6 vertices . how many faces does this polyhedron have ?

★ Formula Used :-

$\begin{gathered}\\\;\boxed{\sf{\bf{Euler's \: formula}}}\end{gathered}$

${\large{\bold{\sf{\underline{SOLUTION:-}}}}}$

Here Given,

» E = 10

» V = 6

$\longrightarrow \tt F + V - E = 2$

$\longrightarrow \tt F + 6 - 10 = 2$

$\Longrightarrow \tt F = 2+ 10 - 6 = 6$

$\begin{gathered}\\\;\underline{\boxed{\tt{\odot\;\;Hence,\;\;polyhedron \: has\;=\;\bf{\blue{6 \: faces}}}}}\end{gathered}$

____________________________________________

${\large{\bold{\sf{\underline{2. QUESTION:-}}}}}$

A polyhedron has 6 faces and 8 vertices .how many edges does this polyhedron have .

★ Formula Used :-

$\begin{gathered}\\\;\boxed{\sf{\bf{Euler's \: formula}}}\end{gathered}$

${\large{\bold{\sf{\underline{SOLUTION:-}}}}}$

Here Given,

» F = 6

» V = 8

$\longrightarrow \tt F + V - E = 2$

$\longrightarrow \tt 6 + 8 - E = 2$

$\Longrightarrow \tt E = 6+ 8 - 2 = 12$

$\begin{gathered}\\\;\underline{\boxed{\tt{\odot\;\;Hence,\;\;polyhedron \: has\;=\;\bf{\blue{12\: edges}}}}}\end{gathered}$

____________________________________________

${\large{\bold{\sf{\underline{3. QUESTION:-}}}}}$

A polyhedron has 40 faces and 60 edges . how many vertices does it have ?

★ Formula Used :-

$\begin{gathered}\\\;\boxed{\sf{\bf{Euler's \: formula}}}\end{gathered}$

${\large{\bold{\sf{\underline{SOLUTION:-}}}}}$

Here Given,

» F = 40

» E = 60

$\longrightarrow \tt F + V - E = 2$

$\longrightarrow \tt 40 + V - 60 = 2$

$\Longrightarrow \tt V = 2+ 60 - 40 = 22$

$\begin{gathered}\\\;\underline{\boxed{\tt{\odot\;\;Hence,\;\;polyhedron \: has\;=\;\bf{\blue{22\: vertices}}}}}\end{gathered}$

____________________________________________

${\large{\bold{\sf{\underline{4. QUESTION:-}}}}}$

How many vertices , faces and edges does a hexagon prism have ?

${\large{\bold{\sf{\underline{SOLUTION:-}}}}}$

In A Hexagonal Prism :

No. of vertices = 2×(no. of sides of the polygon)

» No. of vertices = 2×6 = 12

No. of faces = (no. of sides of the polygon) +2

» No. of faces = 6 + 2 = 8

No. of edges = V + F - 2

» No. of edges = 12 + 8 - 2 = 18

$\begin{gathered}\\\;\underline{\boxed{\tt{\odot\;\;Hence,\;\;hexagonal \: prism \: has\;=\;\bf{\blue{12\: vertices .8 \: faces.18 \: edges}}}}}\end{gathered}$

____________________________________________

${\large{\bold{\sf{\underline{5. QUESTION:-}}}}}$

How many vertices , faces and edges does a octagonal pyramid have ?

${\large{\bold{\sf{\underline{SOLUTION:-}}}}}$

In A Octagonal Prism :

No. of vertices = (no. of sides of the polygon)+1

» No. of vertices = 8 + 1 = 9

No. of faces = (no. of sides of the polygon) + 1

» No. of faces = 8 + 1 = 9

No. of edges = V + F - 2

» No. of edges = 9 + 9 - 2 = 16

$\begin{gathered}\\\;\underline{\boxed{\tt{\odot\;\;Hence,\;\;octagonal \: prism \: has\;=\;\bf{\blue{9\: vertices .9 \: faces.16 \: edges}}}}}\end{gathered}$

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${\large{\bold{\sf{\underline{KNOW \: MORE:-}}}}}$

WHAT IS EULER'S FORMULA ?

• Leonard Euler (1707-1783) was a famous Swiss mathematician who discovered a very important relationship among the number of faces, number of edges and number of vertices of a polyhedron.

• In a 3-dimensional figure, let the number of faces be F, the number of edges be E and the number of vertices be V.

★ Then, the Euler's formula is F+ V - E = 2