# Polyhedron: Definition, Types, Euler’s Formula & Solved Examples (2023)

A polyhedron is a figure formed by polygons which enclose a region of 3 -dimensional space. The polygons are called faces, the line segments in which they intersect are called edges, and the endpoints of the edges are called vertices.

In this article, we will learn all about polyhedrons, their parts like faces, edges and vertices, formulas, and types of polyhedrons with solved examples.

## Polyhedron

The Greek words poly, which means numerous, and hedron, which means surface, combine to form the word “polyhedron.” The number of faces of a polyhedron determines what type it is.

A polyhedron is a closed solid with plane faces enclosing it. A polyhedron’s faces are all polygons. A cube is a polyhedron with six right-angled polygonal edges. There are only five conceivable regular polyhedrons that have congruent faces, each a regular polygon and meeting at equal angles, despite the fact that regular polygons can have any number of sides.

The five regular polyhedrons are also known as the Platonic solids, although they were known to the Greeks before the time of Plato. They are the tetrahedron, bounded by four equilateral triangles; the hexahedron, or cube, bounded by six squares; the octahedron, bounded by eight equilateral triangles; the dodecahedron, bounded by twelve regular pentagons; and the icosahedron, bounded by twenty equilateral triangles.

### Examples of Polyhedrons

Here are some examples of polyhedron that we see in daily life.

 Examples Images Houses/Buildings: Due to their flat surfaces, the majority of the houses in any area are polyhedrons. A prism: Three rectangles are put together to form a prism, and their bases are triangular. A football or a soccer ball. Soccer balls are made by joining 12 pentagons and 20 hexagons. Pyramids: Tetrahedra, sometimes referred to as a pyramid, are polyhedra with four sides. They are identical to the Egyptian pyramids, which are composed on a base and a variety of triangle faces. All vertices of the base are connected to the same point of intersection. Cubes: Six similar squares are used to create these shapes. The six-sided dice from a board game have this geometric form on them. Structural elements such as beams with a square base. This element is a parallelepiped since it is a solid shape formed by two regular squares and four equal rectangles. Bees build their honeycombs in the shape of hexagonal prisms.

## Parts of a Polyhedron

Every polyhedron has three parts:

Face: the flat surfaces that make up a polyhedron are called its faces. These faces are regular polygons.

Edge: the regions where the two flat surfaces meet to form a line segment are known as the edges.

Vertex: It is the point of intersection of the edges of the polyhedron. A vertex is also known as the corner of a polyhedron. The plural of vertex is called vertices.

## Polyhedron Formula

Swiss mathematician Leonhard Euler demonstrated this for any straightforward polyhedron in the 18th century. Leonhard Euler formulated his polyhedron theorem in the year 1750. The link between the quantity of faces, vertices (corner points), and edges in a convex polyhedron is shown by the theorem.

The well-known Euler’s formula also defines a constant that is unaffected by the rotations and translations of the aforementioned polyhedrons. He draws the conclusion that there are only five regular solids possible and establishes a number of relationships in the proposition.

A polyhedron containing no holes, the sum of the number of vertices V and the number of faces F is equal to the number of edges E plus 2, or V + F=E + 2.

Here is the proof of Euler’s formula for a few polyhedrons.

 Polyhedron Name Shape VerticesV EdgesE FacesF Euler’s FormulaV – E + F = 2 Tetrahedron 4 6 4 4 – 6 + 4 = –2 + 4 = 2 Hexahedron or Cube 8 12 6 8 – 12 + 6 = – 4 + 6 = 2 Octahedron 6 12 8 6 – 12 + 8 = – 6 + 8 = 2 Icosahedron 12 30 20 12 – 30 + 20 = –18 + 20 = 2 Dodecahedron 20 30 12 20 – 30 + 12 = – 10 + 12 = 2.

## Proof of Euler’s Formula

We will use graph theory to prove Euler’s formula.

By viewing the polyhedra as a singly linked planar graph P with V vertices, E edges, and F faces, we are able to establish the formula by applying induction to the edges.

If P has zero number of edges, that is E = 0. The graph will have a single with F = 1. Then, by the Euler’s formula V – E + F = 1 – 0 + 1 = 2. Thus, it satisfies Euler’s formula.

Step 1: Let the Euler’s formula is applicable for a graph with n edges.

(Video) Polyhedrons Eulers Formula and Platonic Solids 3D Shapes

Let P be a graph with n + 1 edges.

Case I: In the event that P is a tree and is does not contain any cycle.. Step 1 makes it simple to demonstrate for trees with any number of edges.

Case II: If P has at least one cycle but is not a tree. Select an edge E1 in P that splits the provided area into two distinct halves, then remove that edge E1, creating a new graph P”.

Faces in P > Faces in P’’

Now, P has n + 1 edges, then P’’ has n edges so by the hypothesis P’’ satisfies the Euler’s formula. For P’’, V’ – E’ + F’ = 2 where V’ = V, E’ = E – 1 and F’ = F – 1.

Now, substituting these values, we get

V’ – E’ + F’ = 2

V – E + 1 + F – 1 = 2

V – E + F = 2

Hence, Euler’s formula is applicable for n + 1 edges.

This proves the Euler’s formula.

## Types of Polyhedron

The names of the polyhedrons are derived from the number of faces they have:

 Name No. of faces Tetrahedron A polyhedron with 4 faces. Pentahedron A polyhedron with 5 faces. Hexahedron A polyhedron with 6 faces. Heptahedron A polyhedron with 7 faces. Octahedron A polyhedron with 8 faces. Nonahedron A polyhedron with 9 faces. Decahedron A polyhedron with 10 faces

Here are some other types of polygons that are very common around us.

Learn about Angles of a parallelogram here.

### Regular polyhedron

Regular polyhedra are uniform and have only one type of congruent regular polygon on each of their faces. Five regular polyhedra are present. The regular polyhedra are also known as the Platonic Solids since they were crucial to Plato’s natural philosophy.

For instance, a cube is a regular polyhedron with squares on all of its faces. Vertex A in the cube below touches the faces ABCD, ABGH, and ADEH. The faces ABCD, BCFG, and ABGH are in contact with vertex B. The faces ABCD, BCFG, and CDEF are touched by the vertex C. Likewise, every cube vertex touches three faces.

These are the only regular polyhedrons.

• Cube
• Tetrahedron
• Octahedron
• Dodecahedron
• Icosahedron

These are the ONLY regular polyhedra. Proof:

Assume there exists a {p, q} regular convex polyhedron. Since every face has p edges there would be a total of pF edges in all except that every edge is shared by two faces. Therefore pF = 2E. On the other hand, q edges meet at every vertex. Since each edge connects two vertices, qV = 2E. Substituting this into the Euler’s formula gives:

2E/p + 2E/q – E = 2 or 1/p + 1/q = 1/2 + 1/E

(Video) Euler's Formula - 3 Dimensional Shapes

First of all, p3 and q3 since a polygon must have at least three vertices and three sides. p and q can’t simultaneously be both greater than 3 because then the left hand side will be at most

1/4 + 1/4 = 1/2 < 1/2 + 1/E. Therefore, either p = 3 or q = 3.

If p = 3, 1/q – 1/6 = 1/E So that q can only be 3, 4 or 5. Solving the equation for E yields E equal 6, 12, or 30, respectively.

Similarly, if r = 3 q can only be 3, 4 or 5 with E equal to 6, 12 and 30 respectively. Then it all comes to five possible pairs. All five actually represent realizable shapes.

 Symbol F E V Name {3,3} 4 6 4 Tetrahedron {3,4} 8 12 6 Octahedron {4,3} 6 12 8 Cube {3,5} 20 30 12 Icosahedron {5,3} 12 30 20 Dodecahedron

### Irregular polyhedron

An irregular polyhedron can have one or more different polygons. A rectangle is an example of this. Any pair of faces could be a different size or even shape than another pair. A square-based pyramid is irregular. True, the four triangles are identical, but the base is in the shape of a square, making it an irregular polyhedron.

Examples of Irregular Polyhedrons

Some more examples of an irregular polyhedron are:

### Convex polyhedron

A convex polyhedron is a polyhedron with the property that for any two points inside the polyhedron, the line segment joining them is contained in the polyhedron. All regular polyhedra (i.e., Platonic solids) are convex.

A convex polyhedron has a finite number of faces (intersections of the convex polyhedron with the supporting hyperplanes). Each face of a convex polyhedron is a convex polyhedron of lower dimension. Faces of the faces are also faces of the original polyhedron. One-dimensional faces are known as edges; zero-dimensional faces are known as vertices. A bounded convex polyhedron is the convex hull of its vertices.

A convex polyhedron is a special case of a convex set. Being an intersection of half-spaces, a convex polyhedron is described by a system of linear inequalities and may be studied by algebraic tools. The methods of minimization of linear forms on a convex polyhedron form the subject of linear programming.

### Concave polyhedron

A polyhedron for which there is at least one plane that contains a face of the polyhedron and that is such that parts of the polyhedron are on both sides of the plane. A concave polyhedron is a polyhedron with the property that there exist two points inside it such that the line segment drawn between them contains points, not in the polyhedron. In other words, a polyhedron is concave exactly when it is not convex.

## Solved Examples on Polyhedron

Now that we have learned about Polyhedrons, here are some solved examples on polyhedrons.

Solved Example 1: Identify the shape whose net is given below.

Solution:

This shape is entirely made of equilateral triangles. When folded, it results in a regular octahedron. Note that since these are all equilateral and congruent faces, it is a regular polyhedron.

Solved Example 2: A polyhedron has 7 faces and 10 vertices. How many edges does the polyhedron have?

Solution:

For any polyhedron,

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

F + V – E = 2

Here, F = 7, V = 10, E = ?

Using above formula,

⇒7 + 10 – E = 2

⇒17 – E = 2

⇒17 – 2 = E

⇒ Ε = 15

Example 3: Find the number of vertices in a polyhedron which has 30 edges and 12 faces.

Solution:

For any polyhedron,

F + V – E = 2

Here, F = 12, V = ?, E = 30

Using above formula,

12 + V – 30 = 2

V – 18 = 2

V = 2 + 18

V = 20

## Polyhedron FAQs

Q.1What does polyhedron mean?

Ans.1 The Greek words poly, which means numerous, and hedron, which means surface, combine to form the word “polyhedron.” The number of faces of a polyhedron determines what type it is. A polyhedron is a closed solid with plane faces enclosing it. A polyhedron’s faces are all polygons.

Q.2How many vertices are in the polyhedron?

(Video) Euler’s Formula for Polyhedra

Ans.2 A polyhedron has multiple vertices.

Q.3What is a coordination polyhedron?

Ans.3 Typically, when we use the word “Coordination Polyhedron,” we are referring to the spatial arrangement or geometrical design of the ligands that are physically connected to the centre atom or ion. Tetrahedral, octahedral, or square planar geometries are reported to be the most prevalent for coordination polyhedra.

Q.4How many bases do polyhedrons have?

Ans.4 A polyhedron has multiple bases.

Q.5Is a cylinder a polyhedron?

Ans.5 A cylinder is not a polyhedron.

Q.6Can a polyhedron have 10 faces, 20 edges and 15 vertices?

Ans.6 According to Euler’s polyhedron formula, Vertices (V) – Edges (E) + Faces (F) = 2.Faces (F) = 10, Vertices (V) = 15, and Edges (E) = 20 in this instance. As a result, 15 – 20 + 10 = 5, which is inconsistent with Euler’s formula. Therefore, a polyhedron cannot have 10 faces, 20 edges, and 15 vertices.

## FAQs

### What is the Euler's formula for a 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.

What is polyhedron examples and definition? ›

A three-dimensional shape with flat polygonal faces, straight edges, and sharp corners or vertices is called a polyhedron. Common examples are cubes, prisms, pyramids. However, cones, and spheres are not polyhedrons since they do not have polygonal faces.

What defines a polyhedron? ›

In geometry, a polyhedron is simply a three-dimensional solid which consists of a collection of polygons, usually joined at their edges. The word derives from the Greek poly (many) plus the Indo-European hedron (seat).

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.

What is Euler's formula with example? ›

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.

How do you solve Euler's method? ›

In Euler's method, you can approximate the curve of the solution by the tangent in each interval (that is, by a sequence of short line segments), at steps of h . In general, if you use small step size, the accuracy of approximation increases.

How many types of polyhedron are there? ›

What are the two types of polyhedrons? The two types of polyhedrons are regular and irregular. Platonic solids come under regular polygons. Irregular polyhedrons include prisms and pyramids.

What is a polyhedron with 5 faces? ›

A pentahedron is polyhedron having five faces. Because there are two pentahedral graphs, there are two convex pentahedra, corresponding to the topologies of the square pyramid and the triangular prism.

What is a polyhedron with 7 faces? ›

A heptahedron is a polyhedron with seven faces. There is a single "regular" heptahedron, consisting of a one-sided surface made from four triangles and three quadrilaterals. It is topologically equivalent to the Roman surface (Wells 1991).

How do you calculate a polyhedron? ›

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. which is what Euler's formula tells us it should be.

### How many faces a polyhedron have? ›

A solid figure consists of four or more plane faces (all polygons), pairs of which meet along an edge, three or more edges meeting at a vertex. Complete step-by-step answer: Polyhedron: A three-dimensional figure whose faces are all polygons. A polyhedron has to have a minimum of four faces.

What is polyhedron figure in math? ›

polyhedron, In Euclidean geometry, a three-dimensional object composed of a finite number of polygonal surfaces (faces). Technically, a polyhedron is the boundary between the interior and exterior of a solid. In general, polyhedrons are named according to number of faces.

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

Can a polyhedron have 7 faces 10 vertices and 12 edges? ›

How many edges does the polyhedron have? Summary: A polyhedron has 7 faces and 10 vertices. The polyhedron has 15 edges.

Can a polyhedron have 7 faces 15 edges and 10 vertices? ›

Hence Euler's Formula is verified, yes it's possible.

What is Euler's method simple explanation? ›

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 Euler's method explain? ›

The Euler method is a first-order method, which means that the local error (error per step) is proportional to the square of the step size, and the global error (error at a given time) is proportional to the step size.

What are the 5 types of regular polyhedrons? ›

The five Platonic solids (regular polyhedra) are the tetrahedron, cube, octahedron, icosahedron, and dodecahedron. The regular polyhedra are three dimensional shapes that maintain a certain level of equality; that is, congruent faces, equal length edges, and equal measure angles.

What is a 3 sided polyhedron called? ›

In geometry, a deltahedron (plural deltahedra) is a polyhedron whose faces are all equilateral triangles. The name is taken from the Greek upper case delta (Δ), which has the shape of an equilateral triangle.

What is a 4 sided polyhedron called? ›

A tetrahedron is a four-sided shape in the polyhedron family.

### What is a 6 faced polyhedron called? ›

A hexahedron is a polyhedron with six faces. The figure above shows a number of named hexahedra, in particular the acute golden rhombohedron, cube, cuboid, hemicube, hemiobelisk, obtuse golden rhombohedron, pentagonal pyramid, pentagonal wedge, tetragonal antiwedge, and triangular dipyramid.

What is a polyhedron with 12 faces called? ›

In geometry, a dodecahedron (Greek δωδεκάεδρον, from δώδεκα dōdeka "twelve" + ἕδρα hédra "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid.

What is a 11 sided polyhedron called? ›

A hendecagon is an 11-sided polygon, also variously known as an undecagon or unidecagon.

What is a 20 sided polyhedron called? ›

The icosahedron – 20-sided polyhedron – is frequent. Most often each face of the die is inscribed with a number in Greek and/or Latin up to the number of faces on the polyhedron.

What is a 9 sided polyhedron called? ›

In geometry, an enneahedron (or nonahedron) is a polyhedron with nine faces. There are 2606 types of convex enneahedron, each having a different pattern of vertex, edge, and face connections. None of them are regular.

What is a 8 sided polyhedron called? ›

The regular octahedron, often simply called "the" octahedron, is the Platonic solid with six polyhedron vertices, 12 polyhedron edges, and eight equivalent equilateral triangular faces, denoted . It is also uniform polyhedron and Wenninger model . It is given by the Schläfli symbol and Wythoff symbol .

What is the simplest polyhedron? ›

Every edge has exactly two faces, and every vertex is surrounded by alternating faces and edges. The smallest polyhedron is the tetrahedron with 4 triangular faces, 6 edges, and 4 vertices.

What is a polyhedron with 13 faces? ›

(polyhedra with 13 faces)

What is the formula for volume of a polyhedron? ›

This formula is often written V = (1/3)Bh, where B is the area of the base and h is the length of the altitude (the height).

Can a polyhedron have 3 triangles only? ›

No, such a polyhedron is not possible. A polyhedron has minimum 4 faces.

### Can a polyhedron have 4 triangles? ›

(iii)Yes, a square pyramid has a square as its base and four triangles as its faces.

What polyhedron has 24 faces? ›

A pentagonal icositetrahedron has 24 (i.e., 20+4) five-sided faces.

What polyhedron has 10 faces? ›

In geometry, a pentagonal trapezohedron or deltohedron is the third in an infinite series of face-transitive polyhedra which are dual polyhedra to the antiprisms. It has ten faces (i.e., it is a decahedron) which are congruent kites.

Which polyhedron has 10 faces and 24 edges? ›

Answer: An octagonal prism has 10 faces, 24 edges, and 16 vertices.

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 9 faces 14 edges and 15 vertices? ›

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

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).

How many faces does a polyhedron with 20 vertices and 30 edges have? ›

How many faces does a polyhedron with 20 vertices and 30 edges? Number of faces of the polyhedron given in the question is 12.

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.

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

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

### What polyhedron has 10 vertices? ›

It is a pentagonal prism.

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

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

What is the formula for any polyhedron? ›

He found that υ − e + f = 2 for every convex polyhedron, where υ, e, and f are the numbers of vertices, edges, and faces of the polyhedron.

What is Euler's formula explain? ›

Euler's Formula for Polyhedrons

Euler's polyhedra formula shows that the number of vertices and faces together is exactly two more than the number of edges. We can write Euler's formula for a polyhedron as: Faces + Vertices = Edges + 2. F + V = E + 2.

Is Euler's formula always 2? ›

It doesn't always add to 2. The reason it didn't work was that this new shape is basically different ... that joined bit in the middle means that two vertices become 1.

What is Euler's rule for 3d shapes? ›

According to Euler's formula for any convex polyhedron, the number of Faces (F) and vertices (V) added together is exactly two more than the number of edges (E).

What are the main types of polyhedron? ›

What are the two types of polyhedrons? The two types of polyhedrons are regular and irregular. Platonic solids come under regular polygons. Irregular polyhedrons include prisms and pyramids.

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.

What is a polyhedron with 12 faces? ›

In geometry, a dodecahedron (Greek δωδεκάεδρον, from δώδεκα dōdeka "twelve" + ἕδρα hédra "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid.

Why is Euler's formula used? ›

Euler's formula relates the complex exponential to the cosine and sine functions. This formula is the most important tool in AC analysis. It is why electrical engineers need to understand complex numbers. Created by Willy McAllister.

### Does Euler's formula work for all polyhedron? ›

Euler's formula is true for the cube and the icosahedron. It turns out, rather beautifully, that it is true for pretty much every polyhedron. The only polyhedra for which it doesn't work are those that have holes running through them like the one shown in the figure below.

Is Euler's formula true for all 3d shapes? ›

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

What is Euler formula for 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.

How do you write Euler's formula in 3 dimensional figure? ›

Euler's formula can be written as F + V = E + 2, where F is equal to the number of faces, V is equal to the number of vertices, and E is equal to the number of edges. Euler's formula states that for many solid shapes the number of faces plus the number of vertices minus the number of vertices is equal to 2.

How do you find the faces of 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.

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