Interior Angles of Polygons

An Interior Angle is an angle within 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° + lx° + 30° = 180°

It works for this triangle


At present tilt a line past 10°:
interior angles triangle 80 70 30
fourscore° + 70° + xxx° = 180°

It withal works!
One angle went up by 10°,
and the other went downwards by x°

Quadrilaterals (Squares, etc)

(A Quadrilateral has 4 straight sides)

Permit's attempt 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 x°:
interior angles 100 90 90 80
80° + 100° + 90° + 90° = 360°

It however adds up to 360°

The Interior Angles of a Quadrilateral add up to 360°

Considering there are 2 triangles in a foursquare ...

interior angles 90 (45,45) 90 (45,45)

The interior angles in a triangle add together upwardly to 180° ...

... and for the square they add up to 360° ...

... because the square can be fabricated from two triangles!

Pentagon

interior angles pentagon

A pentagon has 5 sides, and can be made from three triangles, then you know what ...

... its interior angles add together up to 3 × 180° = 540°

And when it is regular (all angles the same), then each angle is 540° / v = 108°

(Exercise: make certain each triangle here adds up to 180°, and check that the pentagon's interior angles add together upwards 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 together another 180° to the total:

So the full general dominion is:

Sum of Interior Angles = (n−2) × 180°

Each Bending (of a Regular Polygon) = (due north−2) × 180° / north

Maybe an example will assistance:

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

regular decagon

Sum of Interior Angles = (n−ii) × 180°

= (10−2) × 180°

= viii × 180°

= 1440°

And for a Regular Decagon:

Each interior bending = 1440°/10 = 144°

Note: Interior Angles are sometimes called "Internal Angles"