Sum of Angles in a Polygon

A polygon is a geometric figure bounded by edges or sides.
Its name will be designated according to the number of sides it has.
For example, a triangle is a figure that has three sides and a quadrilateral is one that has $4$.
Similarly, a pentagon is a figure that has five sides, a hexagon is one that has six, heptagons, octagons, nonagons or enneagons, and decagons also owe their name to the number of edges or sides that make them up.

In a convex polygon, every segment that connects any two points of the polygon's contour lies solely and exclusively inside the polygon.
In a concave polygon, there will be at least one diagonal segment that connects two points of the polygon and that is entirely outside of it.

In the convex polygon, each and every angle will always be less than 180 degrees, in a concave polygon there will always be at least one angle greater than 180 degrees.

How is the sum of the internal angles of a polygon calculated?
Regardless of the polygon you have in front of you, whether convex or concave, you can always calculate the sum of its internal angles using the following formula:

"The sum of the internal angles of a polygon" $=180\times(n-2)$
knowing that
$n=$ "the number of sides of the polygon" 

Given the following polygon:
How will we discover the sum of its internal angles?
First, we will count how many sides it has.
After counting, we saw that it has $7$ sides.
We will note it as $n=7$
Since n tells us the number of sides,
we will look at the formula that allows us to discover the sum of the internal angles:

$(n-2)\times180=$

and we will apply $n=7$
$(7-2)180=X$
Pay attention! In the formula, there are parentheses that indicate we must first perform the subtraction.
Always make sure to work according to the correct order of mathematical operations to avoid mistakes.

Let's solve the exercise:

$5\times180=900$

The sum of the internal angles of our polygon is $900$.

Observe the following polygon and determine if it is convex or concave.
The polygon is concave. We can draw an external diagonal that connects two points of the polygon.
For example:

Example for calculating the sum of the internal angles of a convex polygon:

The formula is true for any type of polygon, but we want to show you that you can use it in the same way for a convex polygon.

We will review step by step:

1. We will count the number of sides of the polygon and note it as $n$.
2. We will apply the formula.

Solution:
$n=4$
$(4-2)\times 180=$
$2\times 180=360$

The sum of the internal angles of a quadrilateral polygon is $360º$ degrees.

Given a regular polygon, meaning that all its sides and angles are equal to each other, such as a square or equilateral triangle, we can use the formula to calculate the sum of the internal angles and then divide by the number of angles to find out the measure of each one of them.

Exterior angles are those found between one side of the polygon and the extension of the original side. That is: Pay attention to the fact that the exterior angle is located outside of the polygon and hence
its name derives.

The sum of the exterior angles of a polygon will always be $360º$ degrees!

At first glance, it seems to be a strange polygon that will give us difficulty in calculating the sum of its internal angles.

But do not panic!

The formula to calculate the sum of the internal angles of a polygon (of every polygon, even the ones that look weird) is right here and also the steps we must follow.

So, let's get to work!

First, let's count how many sides this polygon has:

Recommendation: Write numbers next to each edge to avoid confusion in the count.

Great! Now we know the number of edges our polygon has: $n=11$

What remains for us to do is to place the data in the formula (with caution and preserving the order of mathematical operations)

$180\left(11-2\right)=$

$180\times 9=1620$

$1620$

is the sum of the internal angles of a polygon with $11$ edges!

Useful information:

all the internal angles of a regular polygon are equal. Therefore, after discovering the sum with the learned formula, you can divide it by the number of angles and find the measure of each one of them.

If you are interested in learning more about other angle topics, you can enter one of the following articles:

• Angle notation
• Sides, vertices, and angles
• Right angle
• Acute angle
• Obtuse angle
• Straight angle
• Adjacent angles
• Vertically opposite angles
• Alternate angles
• Corresponding angles
• Sum and difference of angles

In the blog of Tutorela you will find a variety of articles about mathematics.

Assignment:

Given the square, what is the sum of the angles in the square?

Solution

A square has four angles, each of which is equal to: $90^o$, therefore, the sum of the angles in the square is $360^o$

Answer

$360^o$

Assignment

Given the square, what is the sum of the total angles of the four triangles?

Solution

As mentioned, the sum of the angles in each triangle is $180$

In our case, there are four triangles, so the total amount of the four triangles will be:

$180\times4=720$

Answer

$720$

Assignment

Given the square, what is the value of the sum of the angles $D_1+B+A_1$ ?

Solution

In a square, all angles are equal to: $90^o$.

\( AD \) is a diagonal of the square, and a diagonal of the square is a bisector

Therefore, the angle $D_1$ is equal to: $45^o$

The same is true for the angle $A$ since they are equal.

Therefore, the sum of the angles will be

$45+90+45=$

$90+90=180$

Answer

$180$

Assignment

Determine if it is true or false

In a concave kite, the sum of the angles is $180^o$

Solution

A concave kite is a quadrilateral, and in a quadrilateral the sum of the angles is $360^o$

Answer

False

Assignment

Given the square, what is the value of the sum of the angles $D_1+B$ ?

Solution

In a square, all angles are equal $90^o$

$AD$ is a diagonal in a square and a diagonal in a square is the bisector of an angle

Therefore, the angle $D_1$ is equal to $45^o$

Therefore, the sum of the angles $D_1+B$ is equal to:

$45+90=135$

Answer

$135$