Q1.
If the congruent polygons of a prism have 10 sides, how many faces, edges and vertices does the prism have? What if the polygons have n sides?
Answer
A decagonal prism has 12 faces, 30 edges and 20 vertices.
Faces = 2 decagons + 10 side faces = 12
Vertices = 10 on the top + 10 on the bottom = 20
Edges = 10 (top) + 10 (bottom) + 10 (joining) = 30
Vertices = 10 on the top + 10 on the bottom = 20
Edges = 10 (top) + 10 (bottom) + 10 (joining) = 30
For an n-sided polygon, count the same way:
Faces = n + 2
Edges = 3n
Vertices = 2n
Edges = 3n
Vertices = 2n
Why the edges come in three groups: a prism is two copies of the same polygon, one above the other, joined vertex to vertex. The n sides of the top polygon, the n sides of the bottom polygon, and the n vertical joins are the only edges there are.
Check it yourself: F + V − E = (n + 2) + 2n − 3n = 2 for every n. For a cube (n = 4): 6 + 8 − 12 = 2 ✓. This is Euler’s relation, and it holds for every one of these solids.