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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.