Question

How many faces does a dodecahedron have whose vertices are 20 and edges are 30?
(a) 12
(b) 24
(c) 20
(d) 30

Answer 1

Hint:First write all the conditions given in the question. Then try to relate the terms in Euler’s formula of geometry to the terms in the question. Substitute all the known values to get the unknown value which is the faces of shape. This value of the number of faces is the required result in the question.

Complete step-by-step answer:
The given shape of polyhedron in the question is given as Dodecahedron.
The given number of edges for Dodecahedron is given as 30 edges…..(i)
The given number of vertices for Dodecahedron is given as 20 vertices…..(ii)
Euler’s Formula: Either of the two important mathematical theorems of Euler’s. The first one is topological invariances relating to the number of faces, vertices, and edges of the polyhedron. It is written as F + V = E + 2. This equation can be proved by using the sum of angles. This proof uses the fact that the planar graph formed by polyhedron can be embedded, so all the edges form a straight line segment. Sum up all the angles in each face of a straight line by drawing the graphs. The sum of all the angles in a polygon of k sides is \[\left( k-2 \right)\pi \] and each edge contributes 2 faces. So, we get \[\left( 2E-2F \right)\pi ......\left( iii \right)\]
Now, take it another way, each vertex is surrounded by triangles and its sum leading to a total angle of \[2\pi .\] The vertices outside the face contribute \[\left( 2\pi -\theta \left( V \right) \right),\] where \[\theta \left( V \right)\] denotes the exterior angle of the polygon. The total exterior angles of any polygon is \[\left( 2V-4 \right)\pi .....\left( iv \right)\]
Equating the equations (iii) and (iv), we get,
\[\pi \left( 2E-2F \right)=\left( 2V-4 \right)\pi \]
By simplifying the above equation, we can write it as,
E + 2 = F + V……(v)
where F is the number of faces of the shape, V is the number of vertices of the shape, E is the number of edges of the shape.
By substituting the equation (i) in equation (v), we get,
30 + 2 = F + V
By substituting equation (ii) in the above equation, we get,
30 + 2 = F + 20
By simplifying the equation, we can write as,
F + 20 = 32
By subtracting 20 on both the sides, we can write,
F = 32 – 20 = 12
So, there are 12 faces given to the dodecahedron.
Therefore, option (a) is the right answer.

Note: Be careful with the formula because there are many variables students substitute one value in place of the other. Take care of negative and positive signs. In this case, E + 2 = F + V or E – F = V – 2, both are the same but signs are different. So, take care of which form of the formula you are considering to solve the question.