# Polygons and PolyhedraPolygons

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)}° + ${360 - round(angle(b,a,d).deg) - round(angle(c,b,a).deg) - round(angle(d,c,b).deg)}° =

${round(angle(f,e,i).deg)}° + ${round(angle(g,f,e).deg)}° + ${round(angle(h,g,f).deg)}° + ${round(angle(i,h,g).deg)}° + ${540 - round(angle(f,e,i).deg) - round(angle(g,f,e).deg) - round(angle(h,g,f).deg) - round(angle(i,h,g).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