What is the Significance of Euler’s Formula?

In the world of mathematics, which essentially deals with finding out relations and patterns as they exist in the natural world, there is a specific branch in mathematics called Geometry that tackles the challenge using forms and shapes. In this article, we are going to see a particular relationship called Euler’s Formula that was found by Leonhard Euler, which helped us in many ways to get a grip on a particular set of shapes.  As is the case in arithmetic which deals with an infinite array of numbers, the same is the case with geometry which tries to explain the properties of infinite many possible shapes and forms.

Before diving into the formula found by Euler, which later came to be known as the Euler’s formula, let us have a look into the specific category of shapes and forms wherein the Euler’s formula is applicable that is the polyhedron; you can find more details about these shapes and forms in www.cuemath.com.

Speaking of shapes and forms, no shape is as popular as the pyramids, which have been created thousands of years ago and still are an object of fascination. Now, the pyramid is a good example of a polyhedron as a polyhedron has as many faces just like a pyramid does, and then these faces meet at edges as one can distinctly notice in the many pictures of the pyramid, and then those edges meet at vertices or corners. Such is a polyhedron with faces, edges, and vertices, and Euler found that they are all inter-related.

So, do all the 3-dimensional shapes actually have faces, edges, and vertices, and thus, does Euler’s formula can be used in any shape? Well, let us consider a shape so universal that almost all celestial bodies have this form which is the sphere. In a sphere, there are no edges or no vertices, and hence it is not a polyhedron. Prisms and pyramids are popular examples of a polyhedron.

Now, what is it that Euler noticed about these polyhedrons? Below is the formula he gave:

F + V = E + 2

OR

F + V – E = 2 ( that is a constant)

Here,

• F = Number of faces of the polyhedron
• V = Number of Vertices of the polyhedron
• E = Number of edges of the polyhedron

Now let us consider some common shapes and see how this formula fits.

• Cube: A cube is a 3-Dimensional solid shape, it has 6 sides, and it has 8 vertices and 12 edges.
• So using Euler, we get F + V – E = 6 + 8 – 12 = 14 – 12 = 2
• Cuboid: A cuboid is a polyhedron; it too is like a cube that has six faces or sides, eight vertices, and twelve edges.
• Cone: A cone is a 3-Dimensional shape in geometry that narrows smoothly from a circular base to a point called the vertex.
• So we can imagine it has 0 edges and vertices and 1 face, and since it does not qualify the definition of a polyhedron, that is, it should have faces and edges; thus, it does not follow the Euler formula.
• Cylinder: A cylinder is one of the basic shapes, with two circular bases at a distance. But as the Number of vertices is zero, it does not follow Euler’s formula even though it has 2 edges and 2 faces.
• Sphere: A sphere, probably the most common shape, has zero edges, faces, and vertices; thus, it does not follow Euler’s formula.
• Pyramid: Look at the Great Pyramid of Egypt; the pyramid has a noticeable base, the base can be either square or triangle, and pyramids which have square bases are called square pyramids and pyramids with triangular bases are called triangular pyramids.

In a pyramid, there are six edges and four vertices, and four faces.

So using the Euler formula, we can see 4 + 4 – 6 = 2, so it satisfies the Euler’s formula.