Subject: Geometry

Angles In Polygons

We have already discussed few of the basic things regarding polygons in the previous article. In this one, we will take a closer look on the angular properties of polygons by investigating the angles inside (interior) and outside (exterior) the polygons. This can better help us to demystify some of the few special qualities of polygons and develop a more profound understanding on the unique angular properties of polygons.

Angles inside polygons (interior angles)

Interior angles are angles that are located inside the polygon. The number of interior angles a polygon contain is the same as the number of sides or edges a polygon have. So a triangle has three interior angles since it has three sides or edges. A quadrilateral has four angles, a pentagon has five and so on.


The figure above is a hexagon. It has six interior angles since it has six sides.

Unlike sides or edges, the sum of all interior angles of a given polygon is constant. Thus, whatever the lengths of the sides or edges and how big, irregularly shaped a certain polygon is, if we add up all its interior angles, the answer is the same for a polygon with specific number of sides.

Interior angles of a triangle (or trigon)

A triangle is a polygon with three sides and it has three interior angles. The sum of its interior angles is equal to 180^o.


60^o + 90 ^o + 30^o = 180^o.

Interior angles of a quadrilateral (or tetragon)

The same with triangles, tetragons or quadrilaterals have four interior angles since they have four sides. If you look at the figure below, a tetragon can be divided into two different and non-overlapping triangles. Since a single triangle has a total interior angle of 180^o, then two triangles has a total of 180*2 = 360^o. Therefore, a tetragon's interior angles add up to 360^o.


A tetragon or quadrilateral is made up of two triangles.

90^o+90^o+90^o+90^o = 360^o.

Interior angles of a pentagon

A pentagon is a polygon with five sides, thus it also have 5 interior angles. The same with what we do to tetragons, we will divide the pentagon according to the number of triangles that can be inscribed inside it (see figure below), and found out that there are actually three triangles in a pentagon. Thus, the sum of interior angles in a pentagon is 180*3 = 540^o.

Now, if the pentagon is regular, it has equal measures of interior angles. And since the sum of interior angles is 540^o, then the measure of each interior angle in a regular pentagon is 540 divided by the number of interior angles which is 540/5 = 108^o.

In general...

Notice that as we add a side to a polygon, the number of triangles inside it also increases by one which is tantamount of saying that as we add a side to a polygon, we add another 180^o to the total sum of its interior angle.

Interior Angle of Regular Polygons*

Shape

Nomenclature

Number of Edges or Sides

Sum of Interior Angles

Each Angle (regular polygon only)

triangle or tri-gon

3

180^o

60^o

tetragon or square

4

360^o

90^o

pentagon

5

540^o

108^o

hexagon

6

720^o

120^o

heptagon

7

900^o

128.57^o

octagon

8

1080^o

135^o

nonagon

9

1260^o

140^o

decagon

10

1440^o

144^o

hendecagon

11

1620^o

147.27^o

dodecagon

12

1800^o

150^o

For n-gon...

In general, for an n-gon, the sum of internal angles is given by the following formula,

(n-2)*180^o

And for each interior angle in a regular polygon,

\frac{(n-2)*180^o}{n}

Exterior Angles

Exterior angles in a polygon are angles formed by intersecting sides in a polygon and is usually located outside the polygon except for concave polygons. Usually, exterior angles are supplementary angles to interior angles. Supplementary angles mean that if you add the two angles, you'll get 180^o which is just a straight line.


Interior angle + exterior angle = 180^o.