All you wanted to know about Polygons and Interior Angles

Geometric figures like pentagon, hexagon, octagon, etc. are very intriguing geometric two dimensional figures with their own peculiar properties. In general, we call these figures polygons. A little later you will find the proper definition of polygons.

At times, it might look little scary or daunting to understand the properties of these polygons. If you know polygons in and out, this article may not be for you. It is for learners who would like to understand polygons in a simple common sensical manner.

So let us delve deep into the world of polygons. So, here we go with the most obvious question first –

What is a polygon?

Polygon is any shape made up of 3 or more connecting lines on a flat sheet surface. Hence, polygons are 2-dimensional closed figures made up of straight lines. Thus the intersecting lines have to terminate at the point of intersection.

If you would like to dive deeper into the formal definition and terminology related to polygons, I recommend you check out Chegg’s page on polygons

What is an interior angle of a polygon?

The angles formed in the interior or inside of a polygon where two pair of sides intersect are called interior angles.

interior angle in a polygon

What is a regular polygon?

A polygon in which all sides are equal (equilateral) and all angles are equal (equiangular). Otherwise, it’s an irregular polygon.

How do you find the sum of the interior angles of a polygon?

I can directly give you the formula to calculate the sum of all interior angles of a polygon. But I want you to dive a little deeper and understand how that formula is arrived.

Our goal for today is to figure out how the interior angles of a polygon change as the number of sides of the figure increases. So let us start with the polygon which has smallest number of sides, i.e. triangles.

Triangle

To understand interior angles of a polygon, we have to keep triangles (type of polygon) as the starting point. Triangle has 3 interior angles and 3 sides. It’s a known fact that sum of these 3 interior angles of a triangle is always 180°.

Little out of context, but the moment we think of triangles Pythagoras Theorem is generally the first things which comes to our mind. In a previous post, I have tried exploring, was Pythagoras Theorem actually proved by Pythagoras?

3 sided polygon, sum of interior angles = 180°

Moving forward and looking at polygons with higher number of sides –

Quadrilateral

Quadrilateral is polygon with 4 sides and obviously 4 interior angles.

If we connect the opposite corners of the quadrilateral (i.e. the diagonal) divides the quadrilateral into 2 triangles. If we add the sum of interior angles of theses 2 triangles, we get 180°+180°=360°. Thus the sum of interior angles in a 4 sided polygon is always 360°.

Another way of looking at it: square is a polygon with all sides equal, i.e. it’s a regular polygon. We know in a square each of the 4 angles is equal to 90°. Thus sum of all the interior angles is 90°x4 = 360°

4 sided polygon, sum of interior angles = 360°

Pentagon

Pentagon Polygon

In a 5 sided polygon, also called pentagon, you can join opposite 2 vertices from any one vertex and you will get 3 triangles. Again, we know that sum of interior angles of a triangle is 180°.

Thus, sum of interior angles of 3 triangles = sum of all interior angles of a pentagon = 180°x3 = 540°

5 sided polygon, sum of interior angles = 540°

 

Have you noticed that, as we keep increasing the number of sides in a polygon the interior angles keep increasing.

As we move further, i.e. for each increase in number of sides of a polygon, 180° gets added to the sum of interior angles.

So the general rule or formula is,

Sum of Interior Angles = (n-2) x 180°

 

How to find the interior angles of a polygon?

In a ‘n’ sided polygon, the sum of interior angles of a polygon = (n-2) x 180°

A regular polygon is equi-angular; thus each interior angle will be equal. In a ‘n’ sided or ‘n’ angled polygon,

Each Interior Angle = Sum of Interior Angles / no. of sides = [(n-2) x 180°]/n

For example, in a dodecagon (12 sided figure shown below),

dodecagon 12 sided polygon

Sum of interior angles

= (n-2) x 180°

= (12-2) x 180°

= 10 x 180°

= 1800°

And in a regular dodecagon, each interior angle

= 1800°/12

= 150°

What is an exterior angle? How to find an exterior angle?

An exterior angle in any polygon is an angle formed by one side of the polygon and the extension of an adjacent side of the polygon.

Interior and exterior angles are supplementary angles.

Exterior angle = 180° – Interior angle

Thus, if interior angle is , exterior angle = 180° – X°

Another important point to keep in mind is that,

Sum of all exterior angles = 360°

So, in a regular polygon of n sides, each

Exterior angle = 360°/number of sides in a polygon

Example, in a hexagon, each exterior angle will be 360°/6 = 60°

How to find the number of sides of a polygon, if one interior angle is given?

The easiest way to finding the number of sides, is to first find out the exterior angle.

So let’s say the interior angle given is 144°, how to find the number of sides in the polygon?

First step is to calculate the exterior angle, which in this case = 180° – 144° = 36°

Now since it’s known that sum of all exterior angles is 360°, thus the formula is

Number of sides in a polygon = 360°/exterior angle

In above case, number of sides = 360°/36° = 10 sides. Thus it’s a decagon.

For your further reading, you can check this previous post where I’ve explained the ratio of area and volume derived from ratio of sides.

Objective of QuickerMaths is to make mathematics fun, quick and simple. I would love to hear from you in comments below.

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Was Pythagoras Theorem actually proved by Pythagoras?

With all respect to Pythagoras, the so called Pythagoras’ Theorem was known to the ancient Indians long before the time of Pythagoras. Although Pythagoras introduced his theorem to the Western mathematical and scientific world long after, yet that theorem continues to be known as Pythagoras’ Theorem!

There are several proofs given by Indian mathematicians, everyone of which is much simpler than given by Pythagoras or Euclid. Here, we’re going to discuss few such proofs of the Pythagoras theorem:

First Proof of Pythagoras Theorem

a2 + b2 = c2

Where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides of a right angled triangle.

Here, AE pythagoras theorem= BF = CG = DH = x and ED = HC = BG = FA = y

Now, the square ABCD = square EFGH + the 4 congruent right angled triangles around it.

Therefore, z2 + 4 (1/2*x*y) = (x+y)2 or, z2 = x2 + y2. Hence proved.

Second Proof of Pythagoras Theorem

Here we’ll use the property of similar triangles that the area of similar triangles is proportional to the squares of homologous sides.

Any right triangle can be split into two similar smaller right triangles by drawing a perpendicular from point of the right angle. The similarity can be checked using Angle-Angle-Angle similarity.

It’s also evident from the diagram that: similar triangles

Area (Big) =  Area (Medium) + Area (Small)

Now since the triangles are similar they will have the same area equation.

If the long side is c (5), the middle side b (4), and the small side a (3), our area equation for these triangles is:

 

Area = F * hypotenuse^2   where F is some area factor (6/25 or .24 in this case; the exact number doesn’t matter).Now let’s play with the equation:

Area (Big) = Area (Medium) + Area (Small)

F c^2 = F b^2 + F a^2

Divide by F on both sides and you get:

c^2 = b^2 + a^2

Hence we get the famous Pythagoras theorem.

There are more than 100 ways of proving Pythagoras theorem. Which one is your favorite proof?

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Ratio of Area and Volume

Finding the ratio of areas or volumes given the length of a side of a 2 or 3 dimensional figure was always a time consuming task. With the help of the knowledge you are going to acquire now, this will be a simple and quick task.

In any two dimensional figure, if the corresponding sides are in the ratio a:b, then their areas are in the ratio a2:b2

Two dimensional figures can be any polygon like square, rectangle, rhombus, trapezium, hexagon, etc. It can also be a triangle or a circle. The sides, referred in the statement above, can be length, breadth or even diagonal in case of a polygon. In case of a circle the sides will be represented by radius or diameter or circumference. In triangle it can be sides or height of a triangle. Read More

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Understanding Platonic Solids with Modular Origami

A guest post by Maria Rainier

Understanding Platonic Solids with Modular Origami

Solid geometry is perhaps one of the best mathematical applications of origami, but of course, there are many other ways to use it in improving students’ understanding of math’s processes, concepts, and underpinnings. For anyone who has difficulty with the abstract components of math, origami can help provide both visual aids and the opportunity to arrive at mathematical conclusions through trial and error. It’s an especially effective way to help visual and kinesthetic learners to understand basic geometric concepts. Read More

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