Decahedron — Definition, Formula & Examples
A decahedron is any polyhedron that has exactly 10 faces. Unlike the Platonic solids, a decahedron does not refer to one specific shape — many different polyhedra can have 10 faces, with varying numbers of edges and vertices.
A decahedron is a polyhedron with 10 planar faces. The faces may be of mixed polygon types (triangles, quadrilaterals, pentagons, etc.), and the solid is not uniquely determined by the face count alone. The prefix 'deca-' comes from the Greek word for ten.
Key Formula
Where:
- = Number of vertices of the polyhedron
- = Number of edges of the polyhedron
- = Number of faces, which equals 10 for any decahedron
How It Works
Since 'decahedron' only specifies that a solid has 10 faces, you need additional information to pin down its exact shape. One common example is the pentagonal dipyramid augmented with extra faces, or an octagonal prism (which has 2 octagonal faces and 8 rectangular faces, totaling 10). You can always verify a candidate decahedron by counting its faces and checking that it satisfies Euler's formula: .
Worked Example
Problem: An octagonal prism has 2 octagonal faces and 8 rectangular faces. Verify that it is a decahedron and that it satisfies Euler's formula.
Count faces: The prism has 2 octagonal bases and 8 lateral rectangular faces.
Count vertices and edges: Each octagonal base has 8 vertices, giving 16 total. Each base has 8 edges, and there are 8 lateral edges connecting corresponding vertices.
Check Euler's formula: Substitute into Euler's formula for convex polyhedra.
Answer: The octagonal prism has exactly 10 faces, confirming it is a decahedron, and it satisfies Euler's formula.
Why It Matters
Classifying polyhedra by face count is a standard exercise in solid geometry courses. Recognizing that a name like 'decahedron' describes a family of shapes — not a unique solid — helps you understand the broader taxonomy of polyhedra beyond the five Platonic solids.
Common Mistakes
Mistake: Assuming a decahedron is a single unique shape like a cube or icosahedron.
Correction: The term only specifies 10 faces. Many distinct polyhedra qualify, with different vertex and edge counts. You need more information (face types, symmetry) to identify the exact solid.