Question

Verify Euler’s formula for the given solid and also name the solid.​

Answer 1

HINT:

Use the Euler’s Formula (i.e. F +V = E + 2), where F is the total number of Faces in the given solid, V is the total number of the vertices and E is the total number of edges. Count all the faces, vertices and the edges, and to verify the Left-Hand side of Euler’s formula equal to the Right-Hand side of the formula.

Complete step by step answer:

We know that the

$\begin{gathered}\qquad \longmapsto \frak{\underline{\purple{Euler’s \: formula \: is \: }} } \\\end{gathered} \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \begin{gathered}\qquad \longmapsto {\underline{ \bold{\green{ F +V = E + 2 }}} } \\\end{gathered}$

where,

• F is the total number of Faces in the given solid.
• V is the total number of the vertices.
• E is the total number of edges.

The above given solid is a pyramid whose base is rectangle.

The blue dots in the given figure donates vertices (i.e. corner) of the pyramid.

• So, the vertices of the given solid are A, B, C, D, E.
• Hence, total number of vertices = 5 = V
• Edges of the given solid are AB, BC, CD, AD, DE, AE, BE, CE.
• Hence, total number of edges = 8 = E

Now, the faces of the given solid are

◻ABCD,△ABE,△BCE,△CDE,△ADE

Hence, total number of faces = 5 = F

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

Now, from Euler’s formula, we know that:

F +V = E + 2

By putting the value of F, V, E in the above equation, we will get:

⇉5+5=8+2

⇉10=10

Since, we see that the Left-Hand side of the above equation is equal to the Right-Hand side.

So, Euler’s formula is true for the above given solid.

Hence, verified.

Note:

Students usually make mistakes while counting the number of edges, and faces of 3-D figures, because edges and faces usually overlap over each other, when we plot a solid on a plane and they seem to us like they are a single edge or a single face.

$\huge \bold{@DarkShadow7083 }$

Answer 2

Triangular Prism

F = 5

E = 9

V = 6

Euler's Formula => F + V - E = 2

=》 5 + 6 - 9 = 2

=》 11 - 9 = 2

=》2 = 2

LHS = RHS

Hence , verified

Hope it helps you