Question

A polyhedron has 15 edges and 10 vertices. How many faces does it have?

Answer 1

Hint:Use Euler’s formula to get the answer.
In geometry a polyhedron is a solid object necessarily closed, whose surfaces are made from more than 1 polygonal shape. Here, a polygon is referred to a shape with more than 3 straight lines and angles. And using Euler’s formula we can easily find the number of vertices/edges/faces of a polygon. And the formula is given by :-
$V+F-E=2$
where,
V= number of vertices in a polygon
F= number of faces in a polygon
E= number of edges in a polygon
Putting the respective values in this formula would yield the answer we need.

Complete step by step solution:
To solve this particular problem set we will take use of Euler’s formula which states that:-
If a polygon has ‘V’ vertices, ‘F’ faces and ‘E’ edges then the relationship between these 3 numbers can be described by the below formula:-
$V+F-E=2$
where,
V= number of vertices in a polygon
F= number of faces in a polygon
E= number of edges in a polygon
Now, from the given question we will put the respective values and solve the question
E=15 and V=10
So, the equation becomes
$$\begin{gathered} V+F-E=2 \\ \Rightarrow 10+F-15=2 \\ \Rightarrow -5+F=2 \\ \Rightarrow F=5+2 \\ \Rightarrow F=7 \end{gathered}$$
Therefore, the mentioned figure in the question must have 7 polygonal faces.

Note: The values of F,V and E must be kept very carefully and make sure the answer you get is always positive as the number of either Faces, Edges or Vertices can never be negative. Also, the relationship is equal to 2 and not 0 or 1 so make sure you remember the number as this is not any other general equation