How can one define a solid and a three-dimensional solid? If a person knows the faces, vertices, and edges of a solid, they can define a solid. These three are the basic properties to describe any solid, even a three-dimensional solid.
A vertex (plural vertices) is a point where two or more lines, points, or edges meet. Have you come across cones, party hats, edge points of a knife, or anything pointed? All those are vertices.
Cubes and cuboids have eight vertices, cones have a single vertex, cylinders have no vertex, and spheres have no vertex because their surface is curved.
The flat surface on a shape or a solid is known as its face. For example, a cube and cuboid have six faces, cones have a flat surface (at the base) and a curved surface (the lateral part of a cone), cylinders have two flat surfaces, a curved face, and the sphere has all surfaces curved.
When two faces meet, they form a line segment, which is known as the edge. It is also a part of a solid where two vertices meet, or a vertex and a face meet. Cubes and cuboids have twelve edges, cones have only one edge, cylinders have two edges and the sphere has no edge.
Euler was the mathematician who found the relation between vertices, faces, and edges. This relation is valid for flat and closed solids only. This formula is not valid for curved surface objects like cylinders and spheres.
The formula is F + V – E = 2
F, V, and E are the faces, vertices, and edges in an enclosed solid.
For example, a cube has eight vertices, six faces, and twelve edges.
According to the Euler’s formula, F + V – E = 2
This gives, 6 + 8 – 12 = 2
Hence, Euler’s theorem is valid.
This theorem is valid for polyhedrons. What are polyhedrons? A polyhedron is an enclosed figure with edges and flat surfaces. Shapes like pyramids, tetrahedrons, cubes, cuboids, etc., are all polyhedrons.
If all its faces and vertices are congruent to a regular polygon, a (convex) polyhedron (regular convex polyhedron) is formed. Try a simple experiment with sticks. It will allow you to construct five regular polyhedra: tetrahedron, cube, octahedron, icosahedron, and dodecahedron. The Greeks knew these; Plato regarded them as symbolizing the four essential elements: fire, earth, air, and water. He assigned the fifth polyhedron and the dodecahedron to the whole universe. In modern days, these polyhedra are commonly known as Platonic Solids.
Shape | Vertices | Edges | Faces |
Cube | 8 | 12 | 6 |
Cone | 1 | 1 | 2 |
Sphere | 0 | 0 | 1 |
Cylinder | 0 | 3 | 3 |
Rectangular prism | 8 | 12 | 6 |
Triangular prism | 6 | 9 | 5 |
Hexagonal prism | 12 | 18 | 8 |
Pentagonal prism | 10 | 12 | 7 |
Square pyramid | 5 | 8 | 5 |
Octagonal prism | 16 | 24 | 10 |
Triangular pyramid | 4 | 6 | 4 |
Rectangular pyramid | 5 | 8 | 5 |
Pentagonal pyramid | 4 | 10 | 4 |
Hexagonal pyramid | 7 | 12 | 7 |
Octagonal pyramid | 9 | 16 | 9 |