Question

How many edges are there in a cuboid?
(A) $3$
(B) $4$
(C) $5$
(D) $6$
(E) $12$

Answer 1

Hint: We know that a cuboid is a three-dimensional figure having length, breadth and height. The edges of a cuboid are a line segment joining the two adjacent vertices of a cuboid. By Euler’s formula the number of faces $F$, the number of vertices $V$ and the number of edges $E$ of any convex polyhedron are related by the formula\[F + V = E + 2\]. Firstly, we have two find the number of vertices and the number of faces and then by using Euler’s formula we get the number of edges of a cuboid.

Complete step-by-step solution:
In this problem we have to find out how many edges are there in a cuboid. So, let us first draw a cuboid.
Here, the vertices of the cuboid are $A$, $B$,$C$, $D$, $E$ ,$F$ ,$G$ and $H$.
We know that a cuboid is a solid figure which has six rectangular faces perpendicular to each other.
So, there are six faces of a cuboid which are given as $ABCD$, $DCGH$,$ADEH$, $BCGF$,$EFGH$ and $ABFE$.
Now, put the value of $F = 6$ (the number of faces) and the value of $V = 8$ (the number of vertices) in the given Euler’s formula.
$
   \Rightarrow F + V = E + 2 \\
   \Rightarrow 6 + 8 = {\rm E} + 2 \\
   \Rightarrow 14 = {\rm E} + 2 \\
   \Rightarrow {\rm E} = 14 - 2 \\
  \therefore E = 12
 $
Thus, the number of edges of a cuboid are $12$.

Hence, option (E) is the correct answer.

Note: Similarly, we can find the number of edges in a cube or any other polyhedron by using Euler’s formula if the number of edges and the number of faces is given.
The Euler’s theorem is known to be one of the most important mathematical theorems named after Leonhard Euler. Euler’s theorem states a relation between the number of faces, vertices and edges of any polyhedral.