## Polygons

A **polygon**

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

### Angles in Polygons

Every polygon with *n* sides also has *n*

${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

The same also works for larger polygons. We can split a pentagon into

A polygon with *n* sides can be split into

Sum of internal angles in an *n*-gon

### Convex and Concave Polygons

We say that a polygon is **concave***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

Which of these polygons are concave?

### Regular Polygons

We say that a polygon is **regular**

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

angle =

If

### The Area of Regular Polygons

Here you can see a

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

We already know the

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

To find the apothem, we can use the definition of

Now, the area of the isosceles triangle is

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

## 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

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

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

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

For example, every rectangle is also a

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

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

In the previous chapter we showed that

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

### 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?

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

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

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

Now we can use the fact the corresponding parts of congruent triangles are also congruent, to conclude that

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 *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

You might have noticed that all kites are

The diagonal splits the kite into two congruent triangles. We know that they are congruent from the

Using

This means, for example, that the diagonal is a

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

This means that angle α must also be the same as angle β. Since they are adjacent,

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

#### 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

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 width of the rectangle is the distance between the **midsegment** of the trapezium.

Like with

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*:

#### 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

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

*Area* =

#### Rhombus

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 =

*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

**tessellations**

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

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

This has to do with the size of their

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°:

### Tessellations from irregular polygons

We can also try making tessellations out of

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

More surprisingly, *any quadrilateral* also tessellates! Their internal angle sum is

Pentagons are a bit trickier. We already saw that *regular* pentagons

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

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!

### 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 *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:

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**

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

The polygons that make up a polyhedron are called its **faces****edges****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,

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

This equation was discovered by the famous Swiss mathematician

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

In a *regular polyhedron* all **Platonic solids**

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

If we create a polyhedron where three **Tetrahedron** and has *(“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 *(“Octa” means “eight” in Greek. Just like “Octagon” means 8-sided shape, “Octahedron” means 8-faced solid.)*

If **Icosahedron**. It has *(“Icosa” means “twenty” in Greek.)*

If

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

If **cube**. Just like dice, it has *Hexahedron*, after the Greek word “hexa" for “six”.

If

Next, let’s try regular pentagons:

If **Dodecahedron**. It has * (“Dodeca” means “twelve” in Greek.)*

Like before, four or more pentagons

The next regular polygon to try are hexagons:

If three hexagons meet at every vertex, we immediately get a

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

**Tetrahedron**

**Cube**

**Octahedron**

**Dodecahedron**

20 Vertices

30 Edges

**Icosahedron**

12 Vertices

30 Edges

Notice how the number of faces and vertices are **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.

## More on Polyhedra

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

**Archimedean solids**

### 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.

Many **viruses**, **bacteria** and other small **organisms** are shaped like

Many **molecules** are shaped like regular polyhedra. The most famous example is C_{60} which consists of 60 carbon atoms arranged in the shape of a

It was discovered in 1985 when scientists researched interstellar dust. They named it “Buckyball” (or Buckminsterfullerene) after the architect

Most **crystals** have their atoms arranged in a regular grids consisting of

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.

Platonic solids are also used to create **dice**. because of their summetry, every side has the

The

## 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!