# Euler’s Polyhedron Formula (2023)

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In this article, we have explored Euler’s Polyhedron Formula in depth with examples, proof and applications in real life problems.

1. Introduction to Euler’s Polyhedron Formula
2. Examples
3. Euler’s Characteristic
4. Mathematical Proof and converting into Network
5. Application of Euler’s Polyhedron Formula

Euler’s Polyhedron formula states that for all convex Polyhedrons, if we add all the number of faces in a polyhedron, with all the number of polyhedron vertices, and then subtract all the number of polyhedron edges, we always get the number two as a result.

The formula in mathematical terms is as follows-
F+V-E = 2,

where F, V and E stand for the number of faces, vertices and edges in a polyhedron.

To better understand this formula, we need to understand polyhedrons in general.

A polyhedron is a 3d shape that has flat polygonal faces. Lines joining these faces are known as the edges. In addition, we call the corners of these polygonal faces the vertices.

Euler’s formula works for all convex Polyhedrons. For a convex polyhedron, all its diagonals must be inside its structure. Polyhedrons should not have holes in their structure. We consider such polyhedrons as valid applicants for Euler’s formula. In addition, valid Polyhedrons cannot have separate parts, where only one shared edge or vertex exists. However, they are some exclusions as well. Certain non-convex polyhedrons can also produce the same results as Euler’s formula.

### Cube

To further understand Euler’s formula, we can take the example of a cube. (Video) Euler's Formula V - E + F = 2 | Proof

A cube has six faces- top, bottom, left, right, front and back. Counting the corner of all these faces gives us eight vertices. Joining all the faces, we get 12 edges. So according to the formula we get,

6+8-12 = 2

As we can observe from above, adding the number of vertices and faces together, and then subtracting the number edges from the resultant sum, gives us the number two.

### Tetrahedron

We can also consider a Tetrahedron. A Tetrahedron has a triangular face at the bottom and three more triangular faces that share one edge with the bottom edge with a common vertex at the top. This polyhedron has four faces-bottom, front, left and right. In terms of vertices, there is one common vertex shared by three faces at the top. We also have three more vertices, created when the three faces share an edge with the bottom face. This gives us four vertices. In terms of edges, there are three at the bottom and we get three when the side faces join with each other, using a common edge along the side. So, in total, we get six edges.

Applying Euler's formula, we get -

4+4-6 = 2,

which is as per the result we would expect from the formula.

### Octahedron

Another example to consider is an Octahedron. In this Polyhedron, we have four triangular faces, built on top of a square base in the middle. On the bottom of this square base, we also have four more triangular faces below it. So, we can think of the square base in the middle that helps us to join the four top faces with the four bottom faces.

(Video) What is...Euler’s polyhedron formula?

In actuality, the square base is an illusion, that appears when we join the four top faces using one shared edge, at each side, with the bottom four faces. So, we have in total, eight faces.

In terms of vertices, we have one vertex at the top, shared by the top four faces. Likewise, we have one vertex at the bottom. In addition, there are four more vertices in the middle, created when we join the four faces, using an edge for each face, with the bottom four faces. In total, we get six vertices.

In terms of edges, we have four edges, created when the four faces at the top join each other using a shared edge. Likewise, we have four edges for the bottom faces. We also get four more edges, when we join the top four faces with the bottom four faces. In totality, we get 12 edges.

Applying Euler's formula, we get-
8+6 - 12 = 2
This result is satisfactory as per Euler's formula.

### Dodecahedron

We can also look at the case of a Dodecahedron. This Polyhedron contains pentagonal faces. We have one pentagonal face at the bottom, with five more such faces built on top of it. In addition, we then have five more faces that share an edge with the five bottom pentagonal faces. Then, at the top, we have one more pentagonal face. This gives us 12 pentagonal faces in total.

In terms of vertices, we have five vertices at the bottom. As we go to the middle, we get ten more vertices. Considering each pentagonal face, we have five joined with the bottom pentagonal face, and five faces built on top of these five faces. Towards the top, we get five more vertices. This in total, gives us 20 vertices.

When it comes to edges, we have five at the bottom. Then, we get five more edges, as five pentagonal faces built on top of the bottom share one edge with each other, considering from the bottom face. These five faces then share two edges each, with the other five faces built on top of each of them. So, this gives us ten more edges. The pentagonal faces built on top of the five pentagonal faces in the lower half, share one edge with each other. This gives us five more edges. At the top, we have another pentagonal face, which gives us five more edges. So, in total, we have thirty edges.

Applying Euler's formula, we have-
12 + 20 - 30 = 2
This result once again proves Euler's formula.

As mentioned in the introduction, the Euler's formula produces a result of two only in the case of convex polyhedrons. However, there are many 3-d surfaces where the result is not always two, but we can still make use of result from the Euler’s formula. We call the result of the Euler’s formula as Euler’s characteristic, denoted by χ. The formula is shown below.

Χ = V – E + F

(Video) Euler's Formula and Graph Duality

As an extension of the two formulas discussed so far, mathematicians found that the Euler's characteristic for any 3d surface is two minus two times the number of holes present in the surface.

Χ = 2-2g, where g stands for the number of holes in the surface.

We prove Euler's theorem using mathematician Cauchy's method. In this method, we convert the polyhedron into a network. To convert the polyhedron into a network for our proof, we first remove the top face from the cube. Then we flatly lay down each of the remaining faces, ensuring that they still remain connected to each other as shown in the image below. The face that we remove becomes the external face and the rest of the internal faces still count as internal faces. The external face is the area surrounding the network. So, in totality, we still have the same number of faces (including the external face), edges and vertices.

We begin by ensuring that there are no faces with more than three edges. If there is such a face, we divide the face into further triangular faces using a diagonal.
We continue to repeat this step until there are no such faces, as shown below. So, at each step, we have one more face, one more edge and the number of vertices stays the same at every step. Therefore, using Euler’s formula, we get-

F + 1 + V – (E + 1) = F + V + 1 – E – 1 = F + V – E

At each stage of step one, we observe that Euler’s formula always holds till the end. This is shown in the image below. Therefore, the result still holds as per the formula.

(Video) Euler's Formula for Polyhedrons (for 3D Shapes)

In step two, we start removing faces if they share one edge with the external face. We continue to remove such faces, till we no longer have them in our network. We can observe this at each stage in the image below. In the final step, we remove each face that shares two edges with the external face. We continue to do this till all such faces are removed. During this step, we may also repeat step two, but only if in case there are no faces with two shared edges with the external face. We continue doing this, and in the end, get the final triangle as shown below. As observed, in the very end, we get a network that has two faces, one internal and external, three edges and three vertices.

Applying the Euler's formula, we get-
3-3+2,
which gives us the result two as expected.

Euler's formula is significant in graph theory, networking, and computer chip design.

• We use it to check whether a graph is a planar graph. A planar graph is a graph, which we can draw in a plane, in such a way so that no edges cross each other.

• Another application of Euler's formula is to check the connectivity of a graph. Connectivity in a graph requires that a path exists to reach any vertex from any other vertex.

• Also, we use Euler's formula for computer chip designing. Computer chips are much like circuits, containing several components. These chips have tracks that connect all the components in the computer chip. We can think of these circuits as graphs and the tracks as edges. When creating chips, it's vital that these tracks, or edges, don't intersect or cross each other. More generally, we must keep such intersections as low as possible. Thus, we use Euler's formula to find such solutions, where intersections are minimal.

With this article at OpenGenus, you must have the complete idea of Euler’s Polyhedron Formula.

## FAQs

### Why there is always 2 answers to Euler's polyhedral formula? ›

There's a relationship between the number of vertices, faces, and edges on a polyhedron* that will always result in the number '2'. Knowing the relationship results in a formula.

It is written F + V = E + 2, where F is the number of faces, V the number of vertices, and E the number of edges. A cube, for example, has 6 faces, 8 vertices, and 12 edges and satisfies this formula.

What is the Euler's formula for polyhedron? ›

This theorem involves Euler's polyhedral formula (sometimes called Euler's formula). Today we would state this result as: The number of vertices V, faces F, and edges E in a convex 3-dimensional polyhedron, satisfy V + F - E = 2.

Can a polyhedron have 15 faces 30 edges and 20 vertices? ›

Can a polyhedron have 15 faces, 30 edges and 20 vertices? Therefore, a polyhedron cannot have 15 faces, 30 edges and 20 vertices.

Is Euler's formula always 2? ›

There are the same number of edges and faces ... but one less vertex! Oh No! It doesn't always add to 2.

Why does Euler's formula equal 2? ›

V - E + F = 2; or, in words: the number of vertices, minus the number of edges, plus the number of faces, is equal to two.

What is Euler's method for dummies? ›

Euler's Method, is just another technique used to analyze a Differential Equation, which uses the idea of local linearity or linear approximation, where we use small tangent lines over a short distance to approximate the solution to an initial-value problem.

What is a simple explanation of Euler's number? ›

To put it simply, Euler's number is the base of an exponential function whose rate of growth is always proportionate to its present value. The exponential function ex always grows at a rate of ex, a feature that is not true of other bases and one that vastly simplifies the algebra surrounding exponents and logarithms.

Why does Euler's formula work? ›

Euler's Identity stems naturally from interactions of complex numbers which are numbers composed of two pieces: a real number and an imaginary number; an example is 4+3i.

How do you find the number of faces in a polyhedron? ›

Use the fact that V−E+F=2 V − E + F = 2 V-E+F=2 V−E+F=2 to calculate the number of faces for the polyhedron where V V V V is the number of vertices, E E E E is the number of edges and F F F F is the number of faces of the polyhedron.

### Can a polyhedron have 10 faces 28 and 15? ›

Can a polyhedron have 10 faces 20 edges and 15 vertices solve using Euler's formula? ›

Since the Euler's formula does not hold true for the given number of faces, edges and vertices, therefore, there does not exist any polyhedron with 10 faces, 20 edges and 15 vertices.

Can a polyhedron have 10 faces 20 edges and 15 vertices why why not? ›

Q8. Can a polyhedron have 10 faces, 20 edges and 15 vertices? Euler;s formula can't be proved. Hence,a polyhedron can not have 10 faces,20 edges and 15 vertices.

Is Euler's formula true for all polyhedron? ›

Given, Euler's formula is true for all three-dimensional shapes. E for number of edges. Euler's formula is true for the cube and the icosahedron. Therefore, Euler's formula is true only for polyhedra.

What are the exceptions to Euler's formula? ›

There are exceptions to this formula, because it only holds true for a polyhedron that does not intersect itself. Well-known geometrical shapes including spheres, cubes, tetrahedra, and octagons are all non-intersecting polyhedra.

Is Euler's formula accurate? ›

The original proof is based on the Taylor series expansions of the exponential function ez (where z is a complex number) and of sin x and cos x for real numbers x (see below). In fact, the same proof shows that Euler's formula is even valid for all complex numbers x.

What is the most beautiful math equation? ›

Euler's Identity is written simply as: e^(iπ) + 1 = 0, it comprises the five most important mathematical constants, and it is an equation that has been compared to a Shakespearean sonnet. The physicist Richard Feynman called it “the most remarkable formula in mathematics”.

What is the most advanced math equation? ›

For decades, a math puzzle has stumped the smartest mathematicians in the world. x3+y3+z3=k, with k being all the numbers from one to 100, is a Diophantine equation that's sometimes known as "summing of three cubes."

Why is e 2x always positive? ›

Or e2x=(e2)x is a positive function because the base is positive.

Is Euler's method old math? ›

First off, Euler's Method is indeed pretty old, if not exactly ancient. It was developed by Leonhard Euler (pronounced oy-ler), a prolific Swiss mathematician who lived 1707-1783.

### Did NASA use Euler's method? ›

Katherine Johnson (1918 – 2020) used Euler's Method in 1961 at NASA to perform the trajectory analysis that enabled the first human space flight by austronaut Alan Shepard.

Is Euler method step by step method? ›

The Euler method is a first-order method, which means that the local error is proportional to the square of the step size, and the global error is proportional to the step size. The Euler method often serves as the basis to construct more complex methods.

Why is Euler's number so important? ›

It shows up all the time in math and physics, most commonly as a base in logarithmic and exponential functions. It's used to calculate compounding interest, the rate of radioactive decay, and the amount of time it takes to discharge a capacitor.

What is the application of Euler's formula in real life? ›

Euler's method is commonly used in projectile motion including drag, especially to compute the drag force (and thus the drag coefficient) as a function of velocity from experimental data.

Why is Euler's equation the most beautiful theorem in mathematics? ›

Euler's identity is considered to be an exemplar of mathematical beauty as it shows a profound connection between the most fundamental numbers in mathematics. In addition, it is directly used in a proof that π is transcendental, which implies the impossibility of squaring the circle.

Is Euler's method useful? ›

Euler's method is useful because differential equations appear frequently in physics, chemistry, and economics, but usually cannot be solved explicitly, requiring their solutions to be approximated.

What are the pros and cons of Euler's method? ›

Advantages: ➢Euler's method is simple and direct. ➢It can be used for nonlinear IVPs. Disadvantages: ➢It is less accurate and numerically unstable.

What if a polyhedron has 12 edges and 6 faces? ›

A cube has 6 faces, 8 vertices and 12 edges.

What if a polyhedron has 4 faces 6 edges? ›

An tetrahedron has 4 faces and 6 edges.

Can a polyhedron have only 4 faces *? ›

Yes, a polyhedra with exactly 4 triangular faces can exist, A polyhedra with 4 triangular faces is triangular Pyramid. Was this answer helpful?

### Can a polyhedron have 10 faces 20 edges and 1546? ›

No, we cannot have a polyhedron with 10 faces, 20 edges, and 15 vertices as it doesn't satisfy Euler's formula.

Can a polyhedron have 20 faces 30 edges and 12? ›

An icosahedron is made up of triangles. There are 20 triangles in one icosahedron. Above you see a drawing of a icosahedron and a game piece (like dice) in the form of a icosahedron. There are 12 vertices (V = 12), 30 edges (E = 30) and 20 faces (F = 20).

Can a polyhedron have 9 faces 14 edges and 15 vertices? ›

A polyhedron can have 10 faces, 20 edges and 15 vertices.

How to verify Euler's formula of 7 faces 10 vertices and 15 edges? ›

We know, as per Euler's polyhedron formula; V - E + F = 2. Here, V = 10 and F = 7. E = 17 - 2 = 15. Therefore, the polyhedron will have 15 edges.

Can a polyhedron have 20 faces 10 edges and 8 vertices 6? ›

No polyhedron can be formed by these data.

Is it possible to have a polyhedron with 14 faces 24 vertices and 36 edges? ›

The Truncated Octahedron has 6 square faces and 8 regular hexagonal faces which give it 14 Faces, 24 Vertices, and 36 Edges.

What if a polyhedron has 5 faces and 9 edges? ›

The shape which has 6 vertices, 9 edges and 5 faces is called a Triangular prism.

Can a polyhedron have 8 faces 26 edges and 16 vertices? ›

No, a polyhedron cannot have 8 faces, 26 edges and 16 vertices.

Can a polyhedron have 12 faces 30 edges and 5 vertices? ›

Therefore, number of vertices = 20.

Why does sphere have Euler characteristic 2? ›

Since a sphere is homoeomorphic to all regular polyhedrons, the sphere ought to have a Euler Characteristic of 2 as well. A sphere obviously do not have vertices nor edges, which ought to mean they have 2 faces, which i assume are the inside and outside.

### Which polyhedron does not satisfy Euler's formula? ›

Euler formula does not work if the shape has any holes and if the shape is made up of two pieces stuck together(by a vertex or an edge). Example: Imagine two tetrahedrons stuck together by one common vertex. In this case, the number of vertices V = 7, the number of faces F = 8 and the number of edges E = 12.

Why is Euler's method not exact? ›

Euler's Method will only be accurate over small increments and as long as our function does not change too rapidly. Consequently, we need to ensure that our step-size isn't too large or our numerical solution will be inaccurate.

When we learned that the Euler characteristic is equal to 2 under certain conditions what are those conditions? ›

We learned that the Euler characteristic V−E+F is equal to 2 under certain conditions. What are those conditions? The Euler characteristic is 2 as long as there are six or fewer vertices in the graph.

How do you find the Euler number of a polygon? ›

Euler's formula for a simple closed polygon

Suppose that there are T triangles, E edges and V vertices; then Euler's formula for a polygon is T-E+V = 1.

Why is Euler's identity so important? ›

Why Is Euler's Identity Important? Mathematicians love Euler's identity because it is considered a mathematical beauty since it combines five constants of math and three math operations, each occurring only one time. The three operations that it contains are exponentiation, multiplication, and addition.

Is Euler's formula true for sphere? ›

Euler formula:-

It is written F + V - E = 2, where F is the number of faces, V the number of vertices, and E the number of edges. This formula is true for sphere as well.

What is the useless formula for polyhedron? ›

For a polyhedron, the Euler's formula is given by V – E + F = 2 where V is the number of vertices, E is the number edges and F is the number of faces of a polyhedron.

Is Euler's formula true for all shapes? ›

Euler's formula is true for all three-dimensional shapes.

What problem Euler couldn t solve? ›

Euler couldn't solve Fermat's last Theorem. He gave a conjecture of his known which proved to be wrong (that too in the 6th digit). In 1990s Andrew Wiles proved it after solving the Tanayama Shimura Conjecture and Fermat's theorem was a special case of it.

Does Euler's method always overestimate? ›

Since Euler's method is a sequence of tangent line approximations, Euler's method also provides an overestimate, regardless of how many steps are used.

### Which 2 types of forces are only considered in Euler equation? ›

Assumptions of Euler's Equation:

Only pressure and gravity forces are dominant and all other forces are negligible. Flow is steady.

Does Euler's identity prove God? ›

While Leonhard Euler was himself a Christian, it is completely inconceivable that he himself actually thought this was proof of God. It's a non-sequitur, intended to be humorously surprising. Euler knew damn well that it was not really proof of anything at all, much less God.

How do you prove Euler characteristics? ›

Let P be a convex polyhedron, V be the number of vertices of P, E be the number of edges of P, and F be the number of faces of P. Then χ(P) = V − E + F = 2. Remark. This result is known as Euler's Formula.

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