Question
How many faces, edges, and vertices does a triangular prism have? Also verify Euler’s formula for this solid.
Solution — Step by Step
A triangular prism has:
- 2 triangular faces (the top and bottom — called bases)
- 3 rectangular faces (the lateral faces connecting the two triangles)
Total faces
Each triangular face has 3 corners (vertices). Since the two triangles are separate (one at top, one at bottom), they contribute vertices total, with no sharing.
Total vertices
There are three types of edges:
- 3 edges on the top triangular face
- 3 edges on the bottom triangular face
- 3 vertical edges connecting corresponding vertices of the two triangles
Total edges
Euler’s formula for any convex polyhedron states:
Substituting our values:
Euler’s formula is satisfied.
Why This Works
A prism is defined by a polygon at its base — the cross-section is the same all along its length. A triangular prism uses a triangle as its base, giving exactly 2 triangular bases and 3 rectangular lateral faces (one for each side of the triangle).
Euler’s formula holds for all convex polyhedra (closed 3D solids with flat faces). It’s a topological identity — the shape can be stretched or compressed, but as long as it has no holes, the formula holds.
Alternative Method — Quick Counting Rule for Prisms
For a prism with an -sided polygon base:
- Faces =
- Vertices =
- Edges =
For a triangular prism (): , , .
This pattern is worth memorising for Class 8 exams.
Common Mistake
Students often miscount the rectangular faces — some say 4 instead of 3. Remember: the number of rectangular (lateral) faces equals the number of sides of the base polygon. A triangle has 3 sides, so a triangular prism has exactly 3 rectangular faces. If you’re thinking of 4, you might be picturing a cuboid (rectangular prism with a 4-sided base).