Definition

1. Noun:
   • A polyhedron with the property that any plane section through it results in a convex polygon: A convex polyhedron is a three-dimensional geometric shape bounded by flat polygonal faces, where every line segment connecting any two points within the polyhedron lies entirely inside it. This property ensures that any cross-section cut by a plane is a convex polygon.

Usage Examples

• Noun:
  • A cube is a classic example of a convex polyhedron.
  • The algorithm is designed to calculate the volume of any convex polyhedron.
  • In geometry class, we learned to distinguish a convex polyhedron from a concave one.

Advanced Usage

• "to be a convex polyhedron": used to describe the geometric property of a shape.
  • For the mathematical proof, we assumed the object to be a convex polyhedron.
• "faces/vertices/edges of a convex polyhedron": used when discussing the components of the shape.
  • Euler's formula relates the number of faces, vertices, and edges of a convex polyhedron.

Variants and Related Words

• Polyhedron (n): A solid figure with many plane faces, typically more than six. A convex polyhedron is a specific type of polyhedron.
• Convex (adj): Curving or bulging outward. This adjective describes the key property of the polyhedron.
• Convex Hull (n): The smallest convex set that contains a given set of points. The convex hull of a finite set of points in space is a convex polyhedron.
• Platonic Solid (n): A highly symmetric, convex polyhedron whose faces are congruent regular polygons. Examples include the cube and the dodecahedron.

Synonyms

• Convex polytope (in three dimensions): A general term for a convex geometric object with flat sides; in 3D, it is a convex polyhedron.

Related Phrases and Concepts

• Euler's Polyhedron Formula: A theorem stating that for any convex polyhedron, the number of faces (F), vertices (V), and edges (E) satisfy the equation F + V - E = 2.
• Dual Polyhedron: For every convex polyhedron, there exists a dual polyhedron where vertices and faces are swapped.

Noun

1. a polyhedron any plane section of which is a convex polygon