Suppose the length of the side is 6 inches. Consider the right triangle POA. The perimeter of a regular or irregular pentagon is the distance around its five sides. Thus, it is the sum of its sides. Based on angle measures and Pentagon sides, it is categorized into regular and irregular Pentagon, Convex, and Concave Pentagon. The table shows the difference between the pentagons.
Look at the image below to visualize regular and irregular pentagons along with two other types of pentagons - concave and convex pentagons. Example 1: Sara measures the pentagon that she has drawn on the ground. She gets each side as 6 feet, and the apothem is 4 feet long. How will she find the area of the grass patch that she is going to grow? Example 2: Mia decides to make a pentagon shape embroidery in her frock. How much thread will she need to construct a 4 inch sided regular pentagon?
Example 3: Help John to find the apothem of the pentagon of the side measure 16 yards. Example 4: Peter found the perimeter of his pentagonal ground to be units.
How will he find the area of the ground? A two-dimensional shape with 5 sides is known as a pentagon. In other words, we call it a 5-sided polygon. A five-sided shape is called a pentagon. If all five sides are equal then we call it a regular pentagon, whereas if any two of the sides are different in measurement, we call it an irregular pentagon.
On the other hand, a six-sided shape is a hexagon, and an octagon has eight sides. No, a pentagon is not a parallelogram , it is a five-sided polygon. A parallelogram has only four sides. A regular pentagon has 5 lines of symmetry.
There is no line of symmetry for an irregular pentagon. A regular Pentagon will have 0 parallel lines , but an irregular Pentagon can have 2 1 pair or 4 2 pairs parallel lines. A pentagon has five angles. Learn Practice Download. Pentagon Shape A pentagon shape is a flat shape or a flat two-dimensional 5-sided geometric shape. What Is a Pentagon? Formula of Pentagon 3. Pentagon Properties 4. Pentagon Shape Examples 5.
Area of Pentagon 6. It is known that there are 1, distinct fullerenes with 12 pentagons and 20 hexagons, but 1, of them have adjacent pentagons somewhere and are therefore not soccer balls, because they violate condition 2.
The standard soccer ball is the only one with no adjacent pentagons. Leaving behind chemistry and fullerene graphs, let us now consider the crucial question: What other, nonstandard, soccer balls are there, with more than three faces meeting at some vertex, and how can we understand them?
It turns out that we can generate infinite sequences of different soccer balls by a topological construction called a branched coverin g. You can visualize this by imagining the standard soccer-ball pattern superimposed on the surface of the Earth and aligned so that there is one vertex at the North Pole and one vertex at the South Pole.
Now distort the pattern so that one of the zigzag paths along edges from pole to pole straightens out and lies on a meridian, say the prime meridian of zero geographical longitude see Figure 4b. Figure 4. New soccer balls can be made from existing ones by a mathematical construction called a branched covering.
First one chooses a seam of the old soccer ball along the edges of polygons a. This seam is straightened out and sliced open b, c. The whole surface of the soccer ball is shrunk to cover only a hemisphere d. A second copy of this hemisphere is rotated around and stitched to the first e, f. This builds a new soccer ball, which can be deformed as in g. Conforming to the definition of a soccer ball, black faces in the new ball are adjacent to only white faces faces that meet only at vertices are not considered adjacent , and white faces have an alternating sequence of white and black faces around their edges.
Soccer-ball images were calculated and created by Michael Trott using Mathematica. Next, imagine slicing the Earth open along the prime meridian. Shrink the sliced-open coat of the Earth in the east-west direction, holding the poles fixed, until the coat covers exactly half the sphere, say the Western Hemisphere.
Finally, take a copy of this shrunken coat and rotate it around the north-south axis until it covers the Eastern Hemisphere.
Remarkably, the two pieces can be sewn together, giving the sphere a new structure of a soccer ball with twice as many pentagons and hexagons as before. The reason is that at each of the two seams running between the North and South Poles, the two sides of the seam are indistinguishable from the two sides of the cut we made in our original soccer ball.
Therefore, the two pieces fit together perfectly, in such a way that the adjacency conditions 2 and 3 are preserved. See Figure 4 for step-by-step illustrations of this construction. Figure 5. Infinitely many soccer balls can be constructed by the method used in Figure 4.
For example, an eight-fold branched covering of the standard soccer ball can be built by using eight copies of the sliced-open coat of the standard ball to create a soccer ball with 96 pentagons and hexagons. The eight pieces fit together like sections of an orange. The author and his collaborator Volker Braungardt have proved that every soccer ball is a branched covering of the standard one.
The new soccer ball constructed in this way is called a two-fold branched covering of the original one, and the poles are called branch point s. The new ball looks the same as the old one from the topological or rubber-sheet geometry point of view , except at the branch points. Figure 6. The proof that branched coverings produce all soccer balls depends on an analysis of the sequence of colors around any vertex.
Because at least one of the edges meeting at each vertex bounds a pentagon black , there is no vertex where only hexagons white meet. The sequence of faces around a vertex is always black-white-white, black-white-white, and closes up after a number of faces that is a multiple of three. There is a straightforward modification we can make to this construction. Instead of taking two-fold coverings, we can take d -fold branched coverings for any positive integer d.
Instead of shrinking the sphere halfway, we imagine an orange, made up of d orange sections, and for each section we shrink a copy of the coat of the sphere so that it fits precisely over the section. Once again the different pieces fit together along the seams see Figure 5. For all of this it is important that we think of soccer balls as combinatorial or topological—not geometric—objects, so that the polygons can be distorted arbitrarily.
At this point you might think that there could be many more examples of soccer balls, perhaps generated from the standard one by other modifications, or perhaps sporadic examples having no apparent connection to the standard soccer ball.
But this is not the case! Braungardt and I proved that every soccer ball is in fact a suitable branched covering of the standard one possibly with slightly more complicated branching than was discussed above. The proof involved an interesting interplay between the local structure of soccer balls around each vertex and the global structure of branched coverings. Consider any vertex of any soccer ball see Figure 6.
For every face meeting this vertex, there are two consecutive edges that meet there. Because at least one of those two edges bounds a pentagon, by condition 3 , there is no vertex where only hexagons meet.
Thus at every vertex there is a pentagon. Its sides meet hexagons, and the sides of the hexagons alternately meet pentagons and hexagons. This condition can be met only if the faces are ordered around the vertex in the sequence black, white, white, black, white, white, etc. Remember that the pentagons are black. In order for the pattern to close up around the vertex, the number of faces that meet at this vertex must be a multiple of 3. This means that locally, around any vertex, the structure looks just like that of a branched covering of the standard soccer ball around a branch point.
Covering space theory—the part of topology that investigates relations between spaces that look locally alike—then enabled us to prove that any soccer ball is in fact a branched covering of the standard one. To mathematicians, generalization is second nature.
Even after something has been proved, it may not be apparent exactly why it is true. Testing the argument in slightly different situations while probing generalizations is an important part of really understanding it, and seeing which of the assumptions used are essential, and which can be dispensed with.
A quick look at the arguments above reveals that there is very little in the analysis of soccer balls that depends on their being made from pentagons and hexagons. Imagining that we again color the faces black and white, we assume that the black faces have k edges, and the white faces have l edges each.
For conventional soccer balls, k equals 5, and l equals 6. As before, the edges of black faces are required to meet only edges of white faces, and the edges of the white faces alternately meet edges of black and white faces.
The alternation of colors forces l to be an even number. Going one step further in this process of generalization, we can require that every n th edge of a white face meets a black face, and all its other edges meet white faces.
Of course we still require that the edges of black faces meet only white faces. Let us call such a polyhedron a generalized soccer ball. The first question we must ask is: Which combinations of k, m and n are actually possible for a generalized soccer ball?
It turns out that the answer to this question is closely related to the regular polyhedra. Ancient Greek mathematicians and philosophers were fascinated by the regular polyhedra, also known as Platonic solids , attributing to them many mystical properties. The Platonic solids are polyhedra with the greatest possible degree of symmetry: All their faces are equilateral polygons with the same number of sides, and the same number of faces meet at every vertex.
Euclid proved in his Elements that there are only five such polyhedra: the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron see Figure 7.
Figure 7. The five basic Platonic solids shown here have been known since antiquity. Examples of all generalized soccer-ball patterns can be generated by altering Platonic solids.
Although Euclid used the geometric definition of Platonic solids, assuming all the polygons to be regular, modern mathematicians know that the argument does not depend on the geometry. Each Platonic solid can be described by two numbers: the number K of vertices in each face and the number M of faces meeting at each vertex. The possible solutions can be determined quite easily. The complete list of possible values for the pairs K, M is:.
Strictly speaking, this is only the list of genuine polyhedra satisfying the above equation. The equation does have other solutions in positive integers. These solutions correspond to so-called degenerate Platonic solid s, which are not bona fide polyhedra. The first case can be thought of as a beach ball that is a sphere divided into M sections in the manner of a citrus fruit. The Platonic solids give rise to generalized soccer balls by a procedure known as truncation.
Suppose we take a sharp knife and slice off each of the corners of an icosahedron. At each of the 12 vertices of the icosahedron, five faces come together at a point. When we slice off each vertex, we get a small pentagon, with one side bordering each of the faces that used to meet at that vertex.
At the same time, we change the shape of the 20 triangles that make up the faces of the icosahedron. By cutting off the corners of the triangles, we turn them into hexagons. The sides of the hexagons are of two kinds, which occur alternately: the remnants of the sides of the original triangular faces of the icosahedron, and the new sides produced by lopping off the corners. The first kind of side borders another hexagon, and the second kind touches a pentagon.
In fact, the polyhedron we have obtained is nothing but the standard soccer ball. Mathematicians call it the truncated icosahedron. Figure 8. Chopping off corners, or truncation, converts any Platonic solid into a generalized soccer ball. In particular, the standard soccer ball is a truncated icosahedron. After truncation, the 20 triangular faces of the icosahedron become hexagons; the 12 vertices, as shown here, turn into pentagons.
The same truncation procedure can be applied to the other Platonic solids. For example, the truncated tetrahedron consists of triangles and hexagons, such that the sides of the triangles meet only hexagons, while the sides of the hexagons alternately meet triangles and hexagons.
The truncated icosahedron gives values for k, m and n of 5, 3 and 2. Figure 9. Generalized soccer balls fall into 20 types. In this table, k represents the number of sides in any black face; the product m x n is the number of sides in any white face.
Every side of a black face meets a white face. Every n th side of a white face meets a black face. The columns b and w represent the number of black and white faces in the simplest representative of each type.
However, this is not true for other values of n. The minimal realization of type 8 is combinatorially the same as the World Cup ball shown in Figure 2, whereas type 10 is the standard soccer ball. Are these the only possibilities for generalized soccer ball patterns, or are there others?
Just as we did for the Platonic solids, we can express the number of faces, edges and vertices in terms of our basic data. Here this is the number b of black faces, the number w of white faces, and the parameters k , m and n.
Now, because the number of faces meeting at a vertex is not fixed, we do not obtain an equation, but an inequality expressing the fact that the number of faces meeting at each vertex is at least 3. The result is a constraint on k, m and n that can be put in the following form :. This may look complicated, but it can easily be analyzed, just like the equation leading to the Platonic solids.
It is not hard to show that n can be at most equal to 6, because otherwise the left-hand side would be greater than the right-hand side.
With a little more effort, it is possible to compile a complete list of all the possible solutions in integers k, m and n. Alas, the story does not end there. However, Braungardt and I were able to determine the values of k, m, n that do have realizations as soccer balls; these are shown in the table in Figure 9, where we also illustrate the smallest realizations for a few types.
The numbers of hexagons in these examples are 30, 60 and 2, respectively. Note that in the latter case the color scheme is reversed, so the hexagons are black rather than white. The numbers of carbon atoms are 80, and 24, respectively. The last of these is the only fullerene with 24 atoms. In the case of 80 atoms, there are 7 different fullerenes with disjoint pentagons, but only one occurs in our table of generalized soccer balls. For atoms, there are , fullerenes with disjoint pentagons.
Figure Braungardt and I discovered something very intriguing when we tried to see whether every generalized soccer ball comes from a branched covering of one of the entries in our table. However, it is not true for other values of n! The minimal example is just a tetrahedron with one face painted black Figure 10a.
Another realization is an octahedron with two opposite faces painted black Figure 10b. This is not a branched covering of the painted tetrahedron! A branched covering of the tetrahedron would have 3, 6, 9, … faces meeting at every vertex—but the octahedron has 4.
In the tetrahedron example, there are two different kinds of vertices: a vertex at which only white faces meet, and three vertices where one black and two white faces meet. Moreover, the painted octahedron has yet another kind of vertex. Every vertex has the same sequence of colors, which goes black, white, white, black, white, white, …, with only the length of the sequence left open. Because the definition of soccer balls through conditions 1 , 2 and 3 does not specify that soccer-ball polyhedra should be spherical, there is a possibility that they might also exist in other shapes.
Besides the sphere, there are infinitely many other surfaces that might occur: the torus which is the surface of a doughnut , the double torus, the triple torus which is the surface of a pretzel , the quadruple torus, etc. These surfaces are distinguished from one another by their genus , informally known as the number of holes: The sphere has genus zero, the torus has genus one, the double torus has genus two, and so on. Toroidal soccer balls are of two kinds: those that are branched coverings of spherical ones, and those that are not.
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