Polygons and Polyhedra

Geometric shapes are everywhere around us. In this chapter you will learn about angels, polygons, tessellations, polyhedra and nets.

Polygons

A polygon is a closed, flat shape that has only straight sides. Polygons can have any number of sides and angles, but the sides cannot be curved. Which of the shapes below are polygons?

polygon-1
polygon-2
polygon-3
polygon-4
polygon-5
polygon-5_1

We give different names to polygons, depending on how many sides they have:

number-3

Triangle
3 sides

number-4

Quadrilateral
4 sides

number-5

Pentagon
5 sides

number-6

Hexagon
6 sides

number-7

Heptagon
7 sides

number-8

Octagon
8 sides

Angles in Polygons

Every polygon with n sides also has n internal angles. We already know that the sum of the internal angles in a triangle is always ° but what about other polygons?

${round(angle(b,a,d).deg)}° + ${round(angle(c,b,a).deg)}° + ${round(angle(d,c,b).deg)}° + ${round(angle(a,d,c).deg)}°  = 

${round(angle(i,e,f).deg)}° + ${round(angle(e,f,g).deg)}° + ${round(angle(f,g,h).deg)}° + ${round(angle(g,h,i).deg)}° + ${round(angle(h,i,e).deg)}°  = 

It looks like the sum of internal angles in a quadrilateral is always ° – exactly the sum of angles in a triangle. This is no coincidence: every quadrilateral can be split into two triangles.

triangles-4
triangles-1
triangles-2
triangles-3

The same also works for larger polygons. We can split a pentagon into triangles, so its internal angle sum is 3×180°= °. And we can split a hexagon into triangles, so its internal angle sum is 4×180°= °.

A polygon with ${x} sides will have an internal angle sum of 180° × ${x-2} = ${(x-2)*180}°. More generally, a polygon with n sides can be split into triangles. Therefore,

Sum of internal angles in an n-gon =n2×180°.

Convex and Concave Polygons

We say that a polygon is concave if it has a section that “points inwards”. You can imagine that is part has “caved in”. Polygons that are not concave are called convex.

There are two ways you can easily identify concave polygons: they have at least one internal angle that is bigger than 180°. They also have at least one diagonal that lies outside the polygon.

In convex polygons, on the other hand, all internal angles are less than °, and all diagonals lie the polygon.

Which of these polygons are concave?

concave-1
concave-2
concave-3
concave-4
concave-5
concave-6

Regular Polygons

We say that a polygon is regular if all of its sides have the same length, and all of the angles have the same size. Which of these shapes are regular polygons?

regular-1
regular-2
regular-3
regular-4
regular-5
regular-6

Regular polygons can come in many different sizes – but all regular polygons with the same number of sides !

We already know the sum of all internal angles in polygons. For regular polygons all these angles have , so we can work out the size of a single internal angle:

angle = = 180°×x2x=180°360°x.

If n=3 we get the size of the internal angles of an equilateral triangle – we already know that it must be °. In a regular polygon with ${x} sides, every internal angle is 180° – 360°${x} = ${Math.round(180-360/x)}°.

The Area of Regular Polygons

Here you can see a regular polygon with ${n} sides. Every side has length 1m. Let’s try to calculate its area!

First, we can split the polygon into ${toWord(n)} congruent, triangles.

We already know the of these triangles, but we also need the to be able to calculate its area. In regular polygons, this height is sometimes called the apothem.

Notice that there is a right angled triangle formed by the apothem and half the base of the isosceles triangle. This means that we can use trigonometry!

The base angles of the isosceles triangle (let’s call them α) are size of the internal angles of the polygon:

α=12180°360°${n}=${round(90-180/n,2)}°

To find the apothem, we can use the definition of :

tanα=oppositeadjacent=

apothem=12s×tanα=${round(Math.tan(pi/2-pi/n)/2,2)}m

Now, the area of the isosceles triangle is

12base×height=121m×${round(Math.tan(pi/2-pi/n)/2,2)}=${round(Math.tan(pi/2-pi/n)/4,2)}m2

The polygon consists of ${toWord(n)} of these isosceles triangles, all of which have the same area. Therefore, the total area of the polygon is

A=${n}×${round(Math.tan(pi/2-pi/n)/4,2)}=${round(n×Math.tan(pi/2-pi/n)/4,2)}m2

Quadrilaterals

In the previous chapter we investigated many different properties of triangles. Now lets have a look at quadrilaterals.

A regular quadrilateral is called a . All of its sides have the same length, and all of its angles are equal.

A square is a quadrilateral with four equal sides and four equal angles.

For slightly “less regular” quadrilaterals, we have two options. If we just want the angles to be equal, we get a rectangle. If we just want the sides to be equal, we get a rhombus.

A Rectangle is a quadrilateral with four equal angles.

A Rhombus is a quadrilateral with four equal sides.

There are a few other quadrilaters, that are even less regular but still have certain important properties:

If both pairs of opposite sides are parallel, we get a Parallelogram.

If two pairs of adjacent sides have the same length, we get a Kite.

If just one pair of opposite sides is parallel, we get a Trapezium.

Quadrilaterals can fall into multiple of these categories. We can visualise the hierarchy of different types of quadrilaterals as a Venn diagram:

For example, every rectangle is also a , and every is also a kite. A rhombus is a square and a rectangle is a trapezium.

To avoid any ambiguity, we usually use just the most specific type.

Now pick four points, anywhere in the the grey box on the left. We can connect all of them to form a quadrilateral.

Let’s find the midpoint of each of the four sides. If we connect the midpoints, we get .

Try moving the vertices of the outer quadrilateral and observe what happens to the smaller one. It looks like it is not just any quadrilateral, but always a !

But why is that the case? Why should the the result for any quadrilateral always end up being a parallelogram? To help us explain, we need to draw one of the diagonals of the original quadrilateral.

The diagonal splits the quadrilateral into two triangles. And now you can see that two of the sides of the inner quadrilateral are actually of these triangles.

In the previous chapter we showed that midsegments of a triangle are always parallel its base. In this case, it means that both these sides are parallel to the diagonal – therefore they must also be .

We can do exactly the same with the second diagonal of the quadrilateral, to show that both pairs of opposite sides are parallel. And this is all we need to prove that the inner quadrilateral is a parallelogram.

Parallelograms

It turns out that parallelograms have many other interesting properties, other than opposite sides being parallel. Which of the following six statements are true?

The opposite sides are congruent.
The internal angles are always less that 90°.
The diagonals bisect the internal angles.
The opposite angles are congruent.
Both diagonals are congruent.
Adjacent sides have the same length
The two diagonals bisect each other in the middle.

Of course, simply “observing” these properties is not enough. To be sure that they are always true, we need to prove them:

Opposite Sides and Angles

Let’s try to prove that the opposite sides and angles in a parallelogram are always congruent.

Start by drawing one of the diagonals of the parallelogram.

The diagonal creates four new angles with the sides of the of the parallelogram. The two red angles and the two blue angles are alternate angles, so they must each be .

Now if we look at the two triangles created by the diagonal, we see that they have two congruent angles, and one congruent side. By the congruence condition, both triangles must be congruent.

This means that the other corresponding parts of the triangles must also be congruent: in particular, both pairs of opposite sides are congruent, and both pairs of opposite angles are congruent.

It turns out that the converse is also true: if both pairs of opposite sides (or angles) in a quadrilateral are congruent, then the quadrilateral has to be a parallelogram.

Diagonals

Now prove that the two diagonals in a parallelogram bisect each other.

Let’s think about the two yellow triangles generated by the diagonals:

  • We have just proved that the two green sides are congruent, because they are opposite sides of a parallelogram.
  • The two red angles and two blue angles are congruent, because they are .

By the condition, both of the yellow triangles must therefore also be congruent.

Now we can use the fact the corresponding parts of congruent triangles are also congruent, to conclude that AM = CM and BC = DM. In other words, the two diagonals intersect at their midpoints.

Like before, the opposite is also true: if the two diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

Kites

We showed above that the two pairs of sides of a parallelogram are congruent. In a kite, two pairs of adjacent sides are congruent.

The name Kite clearly comes from its shape: it looks like the kites you can fly in the sky. However, of all the special quadrilaterals we have seen so far, the Kite is the only one that can also be concave: if it is shaped like a dart or arrow:

A convex kite

A concave kite that looks like an arrow

You might have noticed that all kites are . The axis of symmetry is .

The diagonal splits the kite into two congruent triangles. We know that they are congruent from the SSS condition: both triangles have three congruent sides (red, green and blue).

Using CPOCT, we therefore know that the corresponding angles must also be congruent.

This means, for example, that the diagonal is a of the two angles at its ends.

We can go even further: if we draw the other diagonal, we get two more, smaller triangles. These must also be congruent, because of the SAS condition: the have the same two sides and included angle.

This means that angle α must also be the same as angle β. Since they are adjacent, supplementary angles both α and β must be °.

In other words, the diagonals of a kite are always .

Area of Quadrilaterals

When calculating the area of triangles in the previous chapter, we used the trick of converting it into a . It turns out that we can also do that for some quadrilaterals:

Parallelogram

On the left, try to draw a rectangle that has the same area as the parallelogram.

Can you see that the missing triangle on the left is the overlapping triangle on the right? Therefore the area of a parallelogram is

Area = base × height

Be careful when measuring the height of a parallelogram: it is usually not the same as the of the two sides.

Trapezium

Recall that trapeziums are quadrilaterals with one pair of parallel sides. These parallel sides are called the bases of the trapezium.

Like before, try to draw a rectangle that has the same area as this trapezium. Can you see how the missing and triangles on the left and the right cancel out?

The height of this rectangle is the the parallel sides of the trapezium.

The width of the rectangle is the distance between the of the two non-parallel sides of the trapezium. This is called the midsegment of the trapezium.

Like with triangles, the midsegment of a trapezium is its two bases. The length of the midsegment is the average of the lengths of the bases: a+c2.

If we combine all of this, we get an equation for the area of a trapezium with parallel sides a and c, and height h:

A=h×a+c2

Kite

In this kite, the two diagonals form the width and the height of a large rectangle that surrounds the kite.

The area of this rectangle is the area of the kite. Can you see how each of the four triangles that make up the kite are the same as the four gaps outside it?

This means that the area of a kite with diagonals d1 and d2 is

Area = 12 d1 × d2.

Rhombus

A Rhombus is a quadrilateral that has four congruent sides. You might remember that every rhombus is a – and also a .

This means that to find the area of a rhombus, we can use either the equation for the area of a parallelogram, or that for the area of a kite:

Area = base × height = 12 d1 × d2.

In different contexts, you might be given different parts of a Rhombus (sides, height, diagonals), and you should pick whichever equation is more convenient.

Tessellations

Polygons appear everywhere in nature. They are especially useful if you want to tile a large area, because you can fit polygons together without any gaps or overlaps. Patterns like that are called tessellations.

honeycomb

Sinaloan Milk Snake skin

Cellular structure of leafs

Basalt columns at Giant’s Causeway in Northern Ireland

Pineapple skin

Shell of a tortoise

Humans have copied many of these natural patterns in art, architecture and technology – from ancient Rome to the present. Here are a few examples:

pavement pattern

Greenhouse at the Eden Project in England

Mosaic at Alhambra

roof at the British Museum in London

Cellular tessellation pavilion in Sydney

Study of Regular Division of the Plane with Reptiles, M. C. Escher

Here you can create your own tessellations using regular polygons. Simply drag new shapes from the sidebar onto the canvas. Which shapes tessellate well? Are there any shapes that don’t tessellate at all? Try to create interesting patterns!

Examples of other students’ tessellations

Tessellations from regular polygons

You might have noticed that some regular polygons (like ) tessellate very easily, while others (like ) don’t seem to tessellate at all.

This has to do with the size of their internal angles, which we learned to calculate before. At every vertex in the tessellation, the internal angles of multiple different polygons meet. We need all of these angles to add up to °, otherwise there will either be a gap or an overlap.

triangles

Triangles because 6 × 60° = 360°.

squares

Squares because 4 × 90° = 360°.

pentagons

Pentagons because multiples of 108° don’t add up to 360°.

hexagons

Hexagons because 3 × 120° = 360°.

You can similarly check that, just like pentagons, any regular polygon with 7 or more sides doesn’t tessellate. This means that the only regular polygons that tessellate are triangles, squares and hexagons!

Of course you could combine different kinds of regular polygons in a tessellation, provided that their internal angles can add up to 360°:

Squares and triangles
90° + 90° + 60° + 60° + 60° = 360°

Squares and triangles
90° + 90° + 60° + 60° + 60° = 360°

Hexagons and triangles
120° + 120° + 60° + 60° = 360°

Hexagons and triangles
120° + 60° + 60° + 60° + 60° = 360°

Hexagons, squares and triangles
120° + 90° + 90° + 60° = 360°

Octagons and squares
135° + 135° + 90° = 360°

Dodecagons (12-gons) and triangles
150° + 150° + 60° = 360°

Dodecagons, hexagons and squares
150° + 120° + 90° = 360°

Tessellations from irregular polygons

We can also try making tessellations out of irregular polygons – as long as we are careful when rotating and arranging them.

It turns out that you can tessellate not just equilateral triangles, but any triangle! Try moving the vertices in this diagram.

The sum of the internal angles in a triangle is °. If we use each angle at every vertex in the tessellation, we get 360°:

More surprisingly, any quadrilateral also tessellates! Their internal angle sum is °, so if we use each angle at every vertex in the tessellation, we we get 360°.

Pentagons are a bit trickier. We already saw that regular pentagons , but what about non-regular ones?

pentagons-1
pentagons-2
pentagons-3

Here are three different examples of tessellations with pentagons. They are not regular, but they are perfectly valid 5-sided polygons.

So far, mathematicians have found 14 different kinds of tessellations with (convex) pentagons. But no one knows if there are others, or if those 14 are all of them. In other words, the problem of finding tessellations of pentagons is still open…

Tessellations in Art

Tessellations we both a tool and inspiration for many artists, architects and designer – most famously the Dutch artist M. C. Escher. Escher’s work contains strange, mutating creatures, patterns and landscapes:

“Sky and Water I” (1938)

“Lizard” (1942)

“Lizard, Fish, Bat” (1952)

“Butterfly” (1948)

“Two Fish” (1942)

“Shells and Starfish” (1941)

These artworks often look fun and effortless, but the underlying mathematical principles are the same as before: angles, rotations, translations and polygons. If the maths isn’t right, the tessellation is not going to work!

“Metamorphosis II” by M. C. Escher (1940)

Penrose Tilings

All the tessellations we saw so far have one thing in common: they are periodic. That means they consist of a regular pattern that is repeated again and again. They can continue forever in all directions and they will look the same everywhere.

In the 1970s, the British mathematician and physicist Roger Penrose discovered non-periodic tessellations – they still continue infinitely in all directions, but never look exactly the same. These are called Penrose tilings, and you only need a few different kinds of polygons to create one:

Move the slider to reveal the underlying structure of this tessellation. Notice how you have the same patterns at various scales: the small yellow pentagons, blue stars, orange rhombi and green ‘ships’ appear in their original size, in a slightly larger size and an even larger size. This self-similarity can be used to prove that this Penrose tiling is non-periodic.

Penrose was exploring tessellations purely for fun, but it turns out that the internal structure of some real materials (like aluminium) follow a similar pattern. The pattern was even used on toilet paper, because the manufacturers noticed that a non-periodic pattern can be rolled up without any bulges.

Polyhedra

Up to now we have just looked at what we can do with polygons in a flat, two-dimensional world. A polyhedron is a 3-dimensional object that is made up of polygons. Here are some examples:

Polyhedra cannot contain curved surfaces – spheres and cylinders, for example, are not polyhedra.

The polygons that make up a polyhedron are called its faces. The lines where two faces are connected are called edges, and the corners where the edges meet are called vertices.

Polyhedra come in many different shapes and sizes – from simple cubes or pyramids with just a few faces, to complex objects like the star above, which has 60 triangular faces. It turns out, however, that all polyhedra have one important property in common:

Euler’s Polyhedron Formula
In every polyhedron, the number of faces (F) plus the number of vertices (V) is two more than the number of edges (E). In other words,

F+V=E+2

For example, if a polyhedron has 12 faces and 18 vertices, we know that it must have edges.

This equation was discovered by the famous Swiss mathematician Leonard Euler. It is true for any polyhedron, as long as it doesn’t contain any holes.

If you play around the different polyhedra, like the ones above, you’ll find that Euler’s formula always works. In a following chapters you’ll learn how to actually prove it mathematically.

Platonic Solids

At the beginning of this chapter, we defined regular polygons as particularly “symmetric” polygons, where all sides and angles are them same. We can do something similar for polyhedra.

In a regular polyhedron all faces are all the same kind of regular polygon, and the same number of faces meet at every vertex. Polyhedra with these two properties are called Platonic solids, named after the Greek philosopher Plato.

So what do the Platonic solids look like – and how many of them are there? To make a 3-dimensional shape, we need at least faces to meet at every vertex. Let’s start systematically with the smallest regular polygon: equilateral triangles:

If we create a polyhedron where three equilateral triangles meet at every vertex, we get the shape on the left. It is called a Tetrahedron and has faces. (“Tetra” means “four” in Greek).

If four equilateral triangles meet at every vertex, we get a different Platonic solid. It is called the Octahedron and has faces. (“Octa” means “eight” in Greek. Just like “Octagon” means 8-sided shape, “Octahedron” means 8-faced solid.)

If triangles meet at every vertex, we get the Icosahedron. It has faces. (“Icosa” means “twenty” in Greek.)

If triangles meet at every vertex, something different happens: we simply get , instead of a 3-dimensional polyhedron.

And seven or more triangles at every vertex also don’t produce new polyhedra: there is not enough space around a vertex, to fit that many triangles.

This means we’ve found Platonic solids consisting of triangles. Let’s move on to the next regular polygon: squares.

If squares meet at every vertex, we get the cube. Just like dice, it has faces. The cube is sometimes also called Hexahedron, after the Greek word “hexa" for “six”.

If squares meet at every vertex, we get . And like before, five or more squares also won’t work.

Next, let’s try regular pentagons:

If pentagons meet at every vertex, we get the Dodecahedron. It has faces. (“Dodeca” means “twelve” in Greek.)

Like before, four or more pentagons because there is not enough space.

The next regular polygon to try are hexagons:

If three hexagons meet at every vertex, we immediately get a . Since there is no space for more than three, it seems like there are no Platonic solids consisting of hexagons.

The same also happens for all regular polygons with more than six sides. They don’t tessellate, and we certainly don’t get any 3-dimensional polygons.

This means that there are just Platonic solids! Let’s have a look at all of them together:

Tetrahedron

Faces
Vertices
Edges

Cube

Faces
Vertices
Edges

Octahedron

Faces
Vertices
Edges

Dodecahedron

Faces
20 Vertices
30 Edges

Icosahedron

Faces
12 Vertices
30 Edges

Notice how the number of faces and vertices are for cube and octahedron, as well as dodecahedron and icosahedron, while the number of edges . These pairs of Platonic solids are called dual solids.

We can turn a polyhedron into its dual, by “replacing” every face with a vertex, and every vertex with a face. These animations show how:

The tetrahedron is dual with itself. Since it has the same number of faces and vertices, swapping them wouldn’t change anything.

Plato believed that all matter in the Universe consists of four elements: Air, Earth, Water and Fire. He thought that every element correspond to one of the Platonic solids, while the fifth one would represent the universe as a whole. Today we know that there are more than 100 different elements which consist of spherical atoms, not polyhedra.

Images from Johannes Kepler’s book “Harmonices Mundi” (1619)

More on Polyhedra

Platonic solids are particularly important polyhedra, but there are countless others.

Archimedean solids, for example, still have to be made up of regular polygons, but you can use multiple different types. They are named after another Greek mathematician, Archimedes of Syracuse, and there are 13 of them:

Truncated Tetrahedron
8 faces, 12 vertices, 18 edges

Cuboctahedron
14 faces, 12 vertices, 24 edges

Truncated Cube
14 faces, 24 vertices, 36 edges

Truncated Octahedron
14 faces, 24 vertices, 36 edges

Rhombicuboctahedron
26 faces, 24 vertices, 48 edges

Truncated Cuboctahedron
26 faces, 48 vertices, 72 edges

Snub Cube
38 faces, 24 vertices, 60 edges

Icosidodecahedron
32 faces, 30 vertices, 60 edges

Truncated Dodecahedron
32 faces, 60 vertices, 90 edges

Truncated Icosahedron
32 faces, 60 vertices, 90 edges

Rhombicosidodecahedron
62 faces, 60 vertices, 120 edges

Truncated Icosidodecahedron
62 faces, 120 vertices, 180 edges

Snub Dodecahedron
92 faces, 60 vertices, 150 edges

Applications of Polyhedra

Plato was wrong in believing that all elements consists of Platonic solids. But regular polyhedra have many special properties that make them appear elsewhere in nature – and we can copy these properties in science and engineering.

Radiolaria skeleton

Icosahedral virus

Many viruses, bacteria and other small organisms are shaped like icosahedra. Viruses, for example, must enclose their genetic material inside a shell of many identical protein units. The icosahedron is the most efficient way to do this, because it consists of a few regular elements but is almost shaped like a sphere.

Buckyball molecule

Montreal Biosphere

Many molecules are shaped like regular polyhedra. The most famous example is C60 which consists of 60 carbon atoms arranged in the shape of a Truncated Icosahedron.

It was discovered in 1985 when scientists researched interstellar dust. They named it “Buckyball” (or Buckminsterfullerene) after the architect Buckminster Fuller, famous for constructing similar-looking buildings.

Fluorite octahedron

Pyrite cube

Most crystals have their atoms arranged in a regular grids consisting of tetrahedra, cubes or octahedra. When they crack or shatter, you can see these shapes on a larger scale.

Octagonal space frames

Louvre museum in Paris

Tetrahedra and octahedra are incredibly rigid and stable, which makes them very useful in construction. Space frames are polygonal structures that can support large roofs and heavy bridges.

Football

Polygonal role-playing dice

Platonic solids are also used to create dice. because of their summetry, every side has the probability of landing facing up – so the dice are fair.

The Truncated Icosahedron is probably the most famous polyhedron in the world: it is the shape of the football.

Nets and Cross Sections

Our entire world is 3-dimensional – but it is often much easier to draw or visualise flat, 2-dimensional objects. And there are a few different ways to view 3-dimensional polyhedra in a 2-dimensional way.

COMING SOON!

Cross-Sections

COMING SOON!