Polygons and PolyhedraPolygons

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?

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

Triangle
3 sides

Quadrilateral
4 sides

Pentagon
5 sides

Hexagon
6 sides

Heptagon
7 sides

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?

${a1[0]}° + ${a1[1]}° + ${a1[2]}° + ${360-a1[0]-a1[1]-a1[2]}° =

${a2[0]}° + ${a2[1]}° + ${a2[2]}° + ${a2[3]}° + ${540-a2[0]-a2[1]-a2[2]-a2[3]}° =

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.

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 =n−2×180°.

Convex and Concave Polygons

We say that a polygon is concave if it has a section that “points inwards”. You can imagine that this 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?

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 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°×x−2x=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} = ${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 the function:

tanα=oppositeadjacent=

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

Now, the area of the isosceles triangle is

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

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Glossary

Polygons and PolyhedraPolygons

Angles in Polygons

Convex and Concave Polygons

Regular Polygons

The Area of Regular Polygons