Pentahedron — Definition, Formula & Examples
A pentahedron is any polyhedron that has exactly five faces. The most common example is a square pyramid, which has four triangular faces and one square base.
A pentahedron is a polyhedron with five planar faces. Two combinatorially distinct types exist: the square pyramid (with 5 vertices and 8 edges) and the triangular prism (with 6 vertices and 9 edges).
How It Works
The prefix "penta-" means five, so a pentahedron is simply a solid bounded by five flat faces. There are only two topologically distinct pentahedra. A square pyramid has a quadrilateral base with four triangular sides meeting at an apex. A triangular prism has two parallel triangular faces connected by three rectangular faces. Both satisfy Euler's formula for polyhedra: .
Worked Example
Problem: A square pyramid has 5 faces and 5 vertices. Verify the number of edges using Euler's formula for polyhedra.
Write Euler's formula: For any convex polyhedron, Euler's formula states:
Substitute known values: The square pyramid has vertices and faces.
Solve for E: Combine the constants and isolate .
Answer: The square pyramid has 8 edges, which matches Euler's formula.
Why It Matters
Classifying polyhedra by face count (tetrahedron, pentahedron, hexahedron, etc.) builds vocabulary you need in solid geometry and standardized math courses. Square pyramids and triangular prisms — both pentahedra — appear frequently in volume and surface area problems.
Common Mistakes
Mistake: Assuming there is only one shape that can be a pentahedron (usually the square pyramid).
Correction: A triangular prism also has exactly five faces (two triangles and three rectangles), so it is equally a pentahedron.