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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.