Polyhedron
Polyhedron
A solid with no curved surfaces or edges. All faces are polygons and all edges are line segments.
See also
Key Formula
V−E+F=2
Where:
- V = Number of vertices (corner points) of the polyhedron
- E = Number of edges (line segments where two faces meet)
- F = Number of faces (flat polygonal surfaces)
Worked Example
Problem: Verify Euler's formula for a cube.
Step 1: Count the vertices. A cube has 8 corner points.
V=8
Step 2: Count the edges. A cube has 12 edges (4 on top, 4 on bottom, 4 vertical).
E=12
Step 3: Count the faces. A cube has 6 square faces.
F=6
Step 4: Substitute into Euler's formula and check that it equals 2.
V−E+F=8−12+6=2✓
Answer: The cube satisfies Euler's formula: 8 − 12 + 6 = 2.
Another Example
This example uses Euler's formula to find a missing count rather than just verifying the formula. It also identifies the specific polyhedron from its face and vertex counts.
Problem: A polyhedron has 5 faces and 6 vertices. Use Euler's formula to find the number of edges.
Step 1: Write down what you know: F = 5 and V = 6.
F=5,V=6
Step 2: Substitute into Euler's formula V − E + F = 2 and solve for E.
6−E+5=2
Step 3: Simplify and solve.
11−E=2⟹E=9
Step 4: Check: this matches a triangular prism (3 rectangular faces + 2 triangular faces = 5 faces, 6 vertices, 9 edges).
6−9+5=2✓
Answer: The polyhedron has 9 edges. It corresponds to a triangular prism.
Frequently Asked Questions
Is a sphere a polyhedron?
No. A sphere has a curved surface, so it is not a polyhedron. Every face of a polyhedron must be a flat polygon, and every edge must be a straight line segment. Since a sphere has no flat faces, straight edges, or vertices, it fails all three requirements.
What is the difference between a polyhedron and a polygon?
A polygon is a flat (two-dimensional) shape with straight sides, such as a triangle or hexagon. A polyhedron is a three-dimensional solid whose faces are polygons. You can think of a polyhedron as being built from polygons joined together in 3D space.
What is Euler's formula for polyhedra?
Euler's formula states that for any convex polyhedron, V − E + F = 2, where V is the number of vertices, E is the number of edges, and F is the number of faces. This relationship holds for all convex polyhedra and many non-convex ones as well, making it a powerful tool for checking your face, edge, and vertex counts.
Polyhedron vs. Prism
| Polyhedron | Prism | |
|---|---|---|
| Definition | Any 3D solid bounded entirely by flat polygonal faces | A specific polyhedron with two identical, parallel polygonal bases connected by rectangular faces |
| Faces | Can be any number ≥ 4 of any polygon types | Always has 2 bases + rectangular lateral faces; total faces = n + 2 for an n-sided base |
| Examples | Cubes, pyramids, prisms, dodecahedra, etc. | Triangular prism, rectangular prism (box), hexagonal prism |
| Relationship | Broad category | A specific type of polyhedron |
Why It Matters
Polyhedra appear throughout geometry courses whenever you study surface area and volume of 3D shapes — every prism, pyramid, and Platonic solid is a polyhedron. Understanding their structure (faces, edges, vertices) and Euler's formula is essential for solving problems on standardized tests and in engineering or architecture contexts. Recognizing whether a solid is a polyhedron also helps you decide which formulas apply to it.
Common Mistakes
Mistake: Counting a cylinder or cone as a polyhedron.
Correction: Cylinders and cones have curved surfaces, so they are not polyhedra. Every surface of a polyhedron must be a flat polygon.
Mistake: Miscounting edges when verifying Euler's formula, especially on complex shapes.
Correction: A systematic approach helps: count edges around each face, then divide by 2 because each edge is shared by exactly two faces. For a shape with faces having a total of S sides, E = S / 2.
Related Terms
- Solid — A polyhedron is a type of solid
- Surface — Flat surfaces form a polyhedron's boundary
- Edge of a Polyhedron — Line segments where two faces meet
- Face of a Polyhedron — Each flat polygonal surface of the solid
- Polygon — 2D shape; faces of a polyhedron are polygons
- Line Segment — Edges of a polyhedron are line segments
- Platonic Solids — The five regular convex polyhedra