Question

How many faces, edges, and vertices do the following have and verify using Euler’s formula
 $$\begin{gathered} \text{A}\text{. pentagonal pyramid} \\ \text{B}\text{. pentagonal prism} \\ \text{C}\text{. square prism} \\ \text{D}\text{. square pyramid} \end{gathered}$$

Answer 1

Hint: Remember the specific shapes of pyramids and prisms.
For a pyramid with a polygon base of n sides:
Vertex =(n+1) Faces= (n+1) Edges= 2n
For a prism with a polygon base of n sides:
Vertex = 2n Faces = (n+2) Edges = 3n
The Euler’s formula is (F+V-E=2).

Complete step-by-step answer:
A) Pentagonal Pyramid

 Faces :6 Vertex: 6 Edges: 10
Verifying Euler’s law: F+V-E=2 ; (6+6-10=2)
Verified.
B) Pentagonal Prism

 Faces :7 Vertex:10 Edges: 15
Verifying Euler’s law: F+V-E=2 ; (7+10-15=2)
Verified.

C) Square Prism

 Faces :6 Vertex: 8 Edges: 12
Verifying Euler’s law: F+V-E=2 ; (6+8-12=2)
Verified.

D) Square Pyramid

 Faces :5 Vertex: 5 Edges: 8
Verifying Euler’s law: F+V-E=2 ; (5+5-8=2)
Verified.

Note: The general formulas could be obtained if one knows the definition and shape of the basic pyramid and prism.
Pyramid: A 3-D figure having a polygon base , and an apex to which all the vertices are connected.
Prism: A 3-D figure, having two polygon faces placed opposite to each other having their vertices connected by edges.