Question

Using Euler’s formula, complete the following table:

	Faces	Vertices	Edges
(i)	$6$		$12$
(ii)		$5$	$8$
(iii)	$14$	$24$
(iv)		$16$	$30$
(v)	$16$		$42$
(vi)	$19$	$19$

Answer 1

Hint: Euler’s formula works for polyhedrons.
Polyhedrons are closed and non-intersecting figures. They are closed solid shapes with flat faces and vertices.

Complete step by step solution:
Leonhard Euler invented Euler’s formula when he started focusing on exponential functions instead of logarithmic functions. He compared series of both exponential and trigonometric functions to derive the Euler’s formula.
Euler’s formula is given as:
$F + V = E + 2$
where, F = Number of faces, V = Number of vertices, E = Number of edges
If any of the two variables out of three (E, F, V) of the above equation are given, the value of the third variable can be determined easily using Euler’s formula.
The different values in the table are calculated as:

	Faces	Vertices	Edges
(i)	$6$	$V = E + 2 - F$ $ =12+2-6 \\ =8 \\ $	$12$
(ii)	$F = E + 2 – V$= $ =8+2-5 \\ =5 \\ $	$5$	$8$
(iii)	$14$	$24$	$E = F + V - 2$$ =14+24-2 \\ =36 \\ $
(iv)	$F = E + 2 - V$$ =30+2-16 \\ =16 \\ $	$16$	$30$
(v)	$16$	$V = E + 2 – F$$ =42+2-16 \\ =28 \\ $	$42$
(vi)	$19$	$19$	E = F + V - $2$$ =19+19-2 \\ =36 \\ $

The vertices, faces and edges at different levels have been determined using Euler’s formula in the above table.
The Euler’s formula also helps in determining the kind of shape formed by a combination of faces, vertices and edges.

Note:
> Euler’s formula was given by mathematician Leonhard Euler.
> Euler’s formula relates faces, vertices and edges of polyhedrons together by formula known as Euler’s formula.