Answer :
To solve this problem, we can use Euler's formula for polyhedra, which states:
[tex]\[ V + F = E + 2 \][/tex]
where:
- [tex]\( V \)[/tex] is the number of vertices
- [tex]\( F \)[/tex] is the number of faces
- [tex]\( E \)[/tex] is the number of edges
For a pyramid, we are given that the number of vertices is equal to the number of faces, so [tex]\( V = F \)[/tex].
We are also given that the pyramid has 40 edges [tex]\((E = 40)\)[/tex].
Substitute the known values into Euler's formula:
[tex]\[ V + F = 40 + 2 \][/tex]
Since [tex]\( V = F \)[/tex], we can express everything in terms of [tex]\( V \)[/tex]:
[tex]\[ V + V = 40 + 2 \][/tex]
[tex]\[ 2V = 42 \][/tex]
To find the number of vertices ([tex]\( V \)[/tex]), divide both sides of the equation by 2:
[tex]\[ V = \frac{42}{2} = 21 \][/tex]
Therefore, the regular pyramid with 40 edges has 21 vertices.
[tex]\[ V + F = E + 2 \][/tex]
where:
- [tex]\( V \)[/tex] is the number of vertices
- [tex]\( F \)[/tex] is the number of faces
- [tex]\( E \)[/tex] is the number of edges
For a pyramid, we are given that the number of vertices is equal to the number of faces, so [tex]\( V = F \)[/tex].
We are also given that the pyramid has 40 edges [tex]\((E = 40)\)[/tex].
Substitute the known values into Euler's formula:
[tex]\[ V + F = 40 + 2 \][/tex]
Since [tex]\( V = F \)[/tex], we can express everything in terms of [tex]\( V \)[/tex]:
[tex]\[ V + V = 40 + 2 \][/tex]
[tex]\[ 2V = 42 \][/tex]
To find the number of vertices ([tex]\( V \)[/tex]), divide both sides of the equation by 2:
[tex]\[ V = \frac{42}{2} = 21 \][/tex]
Therefore, the regular pyramid with 40 edges has 21 vertices.
