Convex Polyhedron — Definition, Formula & Examples

A convex polyhedron is a three-dimensional solid with flat polygonal faces where the entire shape bulges outward — no part of the surface dents inward. If you pick any two points inside or on the surface, the straight line connecting them lies entirely within the solid.

A convex polyhedron is a bounded intersection of finitely many closed half-spaces in $\mathbb{R}^3$ that has nonzero volume. Equivalently, it is a polyhedron $P$ such that for every pair of points $A, B \in P$ , the line segment $\overline{AB} \subseteq P$ .

How It Works

To test whether a polyhedron is convex, examine its vertices. At every vertex, the interior angles of the faces meeting there must sum to less than

360°

. You can also use the line-segment test: choose any two points inside the solid and check that the segment between them never passes outside. All Platonic solids (tetrahedron, cube, octahedron, dodecahedron, icosahedron) are convex. A star-shaped solid or an L-shaped block, by contrast, is not convex because you can find two interior points whose connecting segment exits the solid.

Worked Example

Problem: A cube has 6 faces, 8 vertices, and 12 edges. Verify that it satisfies Euler's formula for convex polyhedra and confirm it is convex.

Step 1: Apply Euler's formula

V - E + F = 2

for convex polyhedra.

8 - 12 + 6 = 2 \checkmark

Step 2: Check convexity at each vertex. Three square faces meet at every vertex of a cube, contributing

3 \times 90° = 270°

, which is less than

360°

3 \times 90° = 270° < 360°

Step 3: Since every vertex satisfies the angle condition and Euler's formula holds, the cube is a convex polyhedron.

Answer: The cube is a convex polyhedron with

V - E + F = 2

and all vertex angle sums equal to

270°

, which is less than

360°

Why It Matters

Convexity is a key requirement in Euler's formula

V - E + F = 2

, which appears throughout high-school geometry. Architecture and computer graphics rely on convex polyhedra for structural stability calculations and efficient 3D rendering algorithms.

Common Mistakes

Mistake: Assuming every polyhedron is convex.

Correction: Many polyhedra — such as star polyhedra or solids with indentations — are non-convex (concave). Always check that no part of the surface curves inward before applying properties that require convexity.