Interior Angles of Polygons and explaination

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Interior Angles of Polygons

An Interior Angle is an angle inside a shape

interior exterior angles
Another example:
interior exterior angles

Triangles

The Interior Angles of a Triangle add up to 180°
Let's try a triangle:
interior angles triangle 90 60 30
90° + 60° + 30° = 180°
It works for this triangle


Now tilt a line by 10°:
interior angles triangle 80 70 30
80° + 70° + 30° = 180°
It still works!
One angle went up by 10°,
and the other went down by 10°

Quadrilaterals (Squares, etc)

(A Quadrilateral has 4 straight sides)
Let's try a square:
interior angles square 90 90 90 90
90° + 90° + 90° + 90° = 360°
A Square adds up to 360°


Now tilt a line by 10°:
interior angles 100 90 90 80
80° + 100° + 90° + 90° = 360°
It still adds up to 360°
The Interior Angles of a Quadrilateral add up to 360°

Because there are 2 triangles in a square ...

interior angles 90 (45,45) 90 (45,45)
The interior angles in a triangle add up to 180° ...
... and for the square they add up to 360° ...
... because the square can be made from two triangles!

Pentagon

interior angles pentagon
A pentagon has 5 sides, and can be made from three triangles, so you know what ...
... its interior angles add up to 3 × 180° = 540°
And when it is regular (all angles the same), then each angle is 540° / 5 = 108°
(Exercise: make sure each triangle here adds up to 180°, and check that the pentagon's interior angles add up to 540°)
The Interior Angles of a Pentagon add up to 540°

The General Rule

Each time we add a side (triangle to quadrilateral, quadrilateral to pentagon, etc), we add another 180° to the total:
If it is a Regular Polygon (all sides are equal, all angles are equal)
ShapeSidesSum of
Interior Angles		ShapeEach Angle
Triangle3180°regular triangle60°
Quadrilateral4360°regular quadrilateral90°
Pentagon5540°pentagon regular108°
Hexagon6720°hexagon regular120°
Heptagon (or Septagon)7900°heptagon refular128.57...°
Octagon81080°octagon regular135°
Nonagon91260°nonagon regular140°
..............
Any Polygonn(n-2) × 180°regular n gon(n-2) × 180° / n
So the general rule is:
Sum of Interior Angles = (n-2) × 180°
Each Angle (of a Regular Polygon) = (n-2) × 180° / n
Perhaps an example will help:

Example: What about a Regular Decagon (10 sides) ?

regular decagon
Sum of Interior Angles= (n-2) × 180°
 = (10-2)×180° = 8×180° = 1440°

And it is a Regular Decagon so:
Each interior angle = 1440°/10 = 144°

Note: Interior Angles are sometimes called "Internal Angles"

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