Question

Verify the Euler formula for the solid:

Answer 1

Hint: First of all, we will mention the Euler formula which needs to be verified in the question. Then, we will individually find the number of edges, faces and whatever is required to put in the formula to verify it.

Step-By-Step answer:
We know that the Euler’s formula is given by the following expression:-
\[ \Rightarrow \] V – E + F = 2; where V refers to the number of vertices in the figure, E refers to the term “number of edges” in the figure and F refers to the “number of faces” in the figure.
The figure given above is definitely a cuboid because it consists of the rectangles as its faces / the coverings.
We know that in a cuboid, 8 vertices are connected using 12 edges which eventually leads to 6 faces.
Therefore, we have V = 8, E = 12 and F = 6.
$ \Rightarrow $ LHS = V – E + F = 8 – 12 + 6
Solving the calculations to get the following:-
$ \Rightarrow $ LHS = V – E + F = - 4 + 6
Solving the calculations further to get the following:-
$ \Rightarrow $ LHS = V – E + F = 2
We can clearly see that this is equal to the RHS.
Hence, LHS = RHS.

$\therefore $ the Euler formula is verified.

Note: The students must note that we can only look at the two of the faces, because we are seeing a side view, 3 of its faces are covered from behind. To help understand this, look at any closed box or carton to imagine and visualize it. It will have 6 faces which are the outer covering of the boxes. Now, count the corners of the box which will come to be equal to 8 and similarly count the joining of these edges which will come to be 12.