What are Vertices?
Vertices (singular: vertex) of any 3D shape can be defined as the points where two or more line segments meet.
What are Edges?
Edges are the line segments around the 3D shape. They are responsible for joining the vertices of any shape.
What are Faces?
Faces are the individual flat surfaces of any 3D shape. Solid figures have more than one face, especially in solids that have polygons as sides.
State the Vertices, Edges and Faces of some Different Shapes
The following table will give an overview of the vertices, edges, and shapes of different solid figures.
What is Euler's formula?
The relation between vertices, edges, and faces can be determined by Euler’s formula, and the formula is written as,
F + V – E = 2, where F, V, and E are the number of faces, vertices, and edges of a solid figure.
Euler’s formula is suitable for closed figures that have flat surfaces and straight edges. This formula cannot be used for cylinders or spheres as they have curved surfaces.
Solved Examples
Example 1: How many faces, edges and vertices are there with reference to a rectangular prism?
Solution: A rectangular prism can be termed as a cuboid as well. It looks like the shape shown in the image.
There are 6 rectangular faces, 12 edges and 8 vertices when we consider a rectangular prism.
Example 2: Use Euler’s formula to prove that a cube has 6 faces, 12 edges and 8 vertices.
Solution: The details provided in the question is to prove that a cube has 6 faces, 12 edges and 8 vertices.
According to Euler’s formula, F + V – E = 2, where F is faces, V is vertices and E is edges.
Let us consider the L.H.S first.
6 + 8 – 12 = 14 – 12 = 2 which is equal to the R.H.S of the formula.
Hence, Euler’s formula is proved for a cube.
Example 3: Mathew is fond of his Rubik’s cube and keeps playing with it. Can you find the number of faces, edges and vertices a Rubik’s cube has?
Solution: As you can see from the image of the Rubik’s cube, it is a 3D figure and as its name suggests, a cube. So, a Rubik’s cube will have 6 faces, 12 edges, and 8 vertices.
A cube has 12 edges.
A triangular prism comprises 5 faces, 6 vertices and 9 edges.
Euler’s formula states that for any solid shape, the number of faces plus the number of vertices minus the number of edges will always be equal to 2. We can write the formula as, F + V − E = 2
A cone has only one edge with one flat surface and one curved surface. The edge is curved for a cone.