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.