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 together upwards to 180°

Let's try a triangle:
interior angles triangle 90 60 30
ninety° + lx° + 30° = 180°

It works for this triangle


Now tilt a line by 10°:
interior angles triangle 80 70 30
lxxx° + lxx° + xxx° = 180°

Information technology still works!
One angle went upward by 10°,
and the other went down by 10°

Quadrilaterals (Squares, etc)

(A Quadrilateral has 4 direct sides)

Allow's try a square:
interior angles square 90 90 90 90
xc° + 90° + xc° + 90° = 360°

A Square adds up to 360°


Now tilt a line past ten°:
interior angles 100 90 90 80
80° + 100° + 90° + xc° = 360°

Information technology still adds upward to 360°

The Interior Angles of a Quadrilateral add upwards to 360°

Because at that place are 2 triangles in a square ...

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

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

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

... because the square can exist made from two triangles!

Pentagon

interior angles pentagon

A pentagon has 5 sides, and tin can be fabricated from three triangles, and so you know what ...

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

And when information technology is regular (all angles the aforementioned), 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 up to 540°)

The Interior Angles of a Pentagon add together up to 540°

The General Rule

Each fourth dimension we add a side (triangle to quadrilateral, quadrilateral to pentagon, etc), we add together another 180° to the total:

So the general rule is:

Sum of Interior Angles = (n−2) × 180°

Each Angle (of a Regular Polygon) = (n−2) × 180° / north

Perhaps an instance volition assist:

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

regular decagon

Sum of Interior Angles = (north−two) × 180°

= (10−two) × 180°

= 8 × 180°

= 1440°

And for a Regular Decagon:

Each interior angle = 1440°/ten = 144°

Annotation: Interior Angles are sometimes called "Internal Angles"