Angles and Polygons - Advanced

To calculate the sum of interior angles in a polygon, we will use the following formula:

When-
$n$ = number of sides in a polygon
The formula works on any polygon, whether it is convex, concave, or regular.

Note - An interior angle is an angle located between 2 sides of the polygon and positioned inside the polygon.

Steps to find the sum of interior angles in a polygon:

1. Let's count how many sides the polygon has.
2. We'll substitute in the formula and get the sum of the polygon's interior angles.

Pay attention -
First, we will perform the operation in parentheses according to the order of operations. We will subtract 2 from the number of sides of the polygon $2$ and then multiply by $180$.

Click here to learn more about the sum of interior angles in a polygon

The sum of exterior angles in a polygon is $360$ degrees in any polygon, regardless of how many sides and angles it has.
First, let's recall what an exterior angle in a polygon is.

An exterior angle is an angle located between an original side and the extension of an original side - a side that extends outside the polygon.
The angle is located outside the polygon and therefore called - an exterior angle.
Let's see this in the illustration:

Look at the original sides of the polygon and imagine that whoever drew the polygon fell asleep in the middle and continued one of its sides a bit too much.
The angle between the original side and the side where they fell asleep while drawing it will be an exterior angle.
The sum of exterior angles will always be $360$ degrees in any polygon you encounter.

Note - An angle between a leg and a leg is not an exterior angle.
For example:

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In a regular polygon, all sides are equal and all interior angles are equal.
To find the measure of an angle in a regular polygon, use the following formula:

when-
$n$ = number of sides in a polygon

Note-
The formula is similar to the formula for the sum of interior angles in a polygon.
What's added is the division by the number of sides of the regular polygon, which equals the number of angles in a regular polygon.

Steps for finding the angle measure of a weighted average:

1. Count how many sides the regular polygon has. It's recommended to write the number next to each side so you don't get confused while counting.
2. Substitute in the formula while maintaining the order of operations and find the measure of the angle in the regular polygon.

Click here to learn more about the angle measures of a regular polygon

Angles in a regular hexagon:

A regular hexagon is a polygon with six sides where all sides are equal and all angles are equal, which makes it a regular polygon.
Therefore, the formula for calculating the interior angle of a hexagon will be identical to the formula for the interior angle in a regular polygon:

Note - in every regular hexagon:
The sum of interior angles is $720$

and the size of each angle will be $120$

And now let's move on to a regular octagon:

A regular octagon is a polygon with eight sides where all sides are equal and all angles are equal.

Therefore, the formula for calculating the interior angle of a hexagon will be identical to the formula for the interior angle of a regular polygon:

Now we will calculate the sum of angles in a regular octagon -

We will use the formula for finding the sum of interior angles in a polygon:
$180*(8-2)=$
$180*6=1080$

And now we'll calculate the value of an angle in a regular octagon:
$1080:8=135$

Since all angles are equal, we simply divided by the number of angles in a regular octagon.

Note - in every regular octagon:
The sum of interior angles is $1080$
and the size of each angle is $1080$ degrees

Click here to learn more about angles in regular hexagons and octagons