25.7 Chapter summary

• Regular polygons are polygons that have all sides equal and all interior angles equal.
• A face is any flat surface on a 3D object.
• An edge is the line segment along which two faces of a 3D object join together.
• A vertex is the corner or point where edges of a 3D object meet.
• The Platonic solids are a special group of 3D objects with faces that are congruent regular polygons.
  • A tetrahedron is an object with four congruent triangular faces.
  • A hexahedron (cube) is an object with six congruent square faces.
  • An octahedron is an object with eight congruent triangular faces.
  • A dodecahedron is an object with twelve congruent pentagonal faces.
  • An icosahedron is an object with twenty congruent triangular faces.
• Euler’s formula states that the difference between the number of vertices (\(v\)) and the number of edges (\(e\)) added to the number of faces (\(f\)) is equal to \(2\): \(v - e + f = 2\)
• General properties of the five Platonic solids:
  • All faces are regular and congruent.
  • The same number of faces meet at each vertex.
  • All the edges are the same length.
  • Euler’s formula is satisfied.
• A net is a flat diagram that can be folded to form a three-dimensional shape.
• A pyramid is an object that has a polygon as its base and triangular sides that converge at an apex.
• The apex is the highest point of an object and it is usually the vertex opposite the base.