Polyhedrons and Their Properties

Polyhedron Definition

• A polyhedron is a solid bounded by planes.
• The intersections of the planes are called edges.
• The intersection of the edges are called vertices.
• The portions of the planes included by the edges are called faces.

Lateral Faces and Edges

• The surfaces on the sides of a polyhedron are called lateral faces.
• The intersections of these faces are called lateral edges.
• A line joining two vertices that are not on the same face is called a diagonal.

Convex and Concave Polyhedrons

• If a plane cuts a polyhedron and the section is a convex polygon, the solid is called a convex polyhedron.
• If the section is a concave polygon, the solid is called a concave polyhedron.

Classification of Polyhedrons

• Polyhedrons are classified by the number of faces.
• Examples include tetrahedron (4 faces), hexahedron (6 faces), octahedron (8 faces), dodecahedron (12 faces), and icosahedron (20 faces).

Types of Polyhedrons

• Prisms: A polyhedron with two parallel and congruent bases, and lateral faces that are parallelograms.
  • Prisms are named based on the shape of their bases (e.g., triangular prism, rectangular prism, pentagonal prism, hexagonal prism, octagonal prism, trapezoidal prism).
• Pyramids: A polyhedron with one base (any polygon) and lateral faces that are triangles, meeting at a common vertex.
• Platonic Solids: A convex polyhedron with all faces being congruent regular polygons.

Prisms

• Triangular prism: 5 faces (2 triangles, 3 rectangles), 9 edges, 6 vertices.
• Rectangular prism: 6 rectangular faces, 12 edges, 8 vertices.
• Square prism: 6 faces (2 squares, 4 rectangles), 12 edges, 8 vertices.
• Pentagonal prism: 7 faces (2 pentagons, 5 rectangles), 15 edges, 10 vertices.
• Hexagonal prism: 8 faces (2 hexagons, 6 rectangles), 18 edges, 12 vertices.
• Octagonal prism: 10 faces (2 octagons, 8 rectangles), 24 edges, 16 vertices.
• Trapezoidal prism: 6 faces (2 trapezoids, 4 rectangles), 12 edges, 8 vertices.

Right and Oblique Prisms

• Right prism: Lateral edges perpendicular to the bases.
• Oblique prism: Lateral edges are not perpendicular to the bases.

Regular Prisms

• A right prism whose bases are regular polygons.

Right Sections

• A section of a prism made by a plane perpendicular to its lateral edges.

Truncated Prisms

• The part of a prism between one base and a section made by a plane oblique to the base, cutting all the lateral edges.

Parallelepiped

• A prism whose bases are parallelograms.
• Right Parallelepiped: A parallelepiped with lateral edges perpendicular to its bases.
• Rectangular Parallelepiped: A right parallelepiped with rectangular faces.
• Cube: A rectangular parallelepiped with six square faces.

Cube Properties

• All faces are congruent squares.
• All three dimensions (edges) are equal.

Cube Formulas

• Area of a face (A) = e²
• Lateral area (S) = 4e²
• Total surface area (T) = 6e²
• Volume (V) = e³

Exercises

• Exercises for practice problems on the cube are given.