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# Interior Angles of Polygons

### An Interior Angle is an angle inside a shape. ## Triangles

### The Interior Angles of a Triangle add up to 180°  ### 80° + 70° + 30° = 180°

It works for this triangle!

Let's tilt a line by 10° ...

It still works, because one angle went up by 10°, but the other went down by 10°

## Quadrilaterals (Squares, etc)

(A Quadrilateral is any shape with 4 sides)  ### 80° + 100° + 90° + 90° = 360°

A Square adds up to 360°

Let's tilt a line by 10° ... still adds up to 360°!

## Because there are Two Triangles in a Square

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

## Pentagon A pentagon has 5 sides, and can be made from three triangles, so you know what ... ... its internal angles add up to 3 × 180° = 540° And if it is a regular pentagon (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 internal angles add up to 540°)

## The General Rule

So, each time we add a side (triangle to quadrilateral, quadrilateral to pentagon, etc), we add another 180° to the total:

Shape Sides Sum of Shape Each Angle Internal Angles If it is a Regular Polygon.. Triangle 3 180° 60° Quadrilateral 4 360° 90° Pentagon 5 540° 108° Hexagon 6 720° 120° ... ... .. ... ... Any Polygon n (n-2) × 180° (n-2) × 180° / n

That last line can be a bit hard to understand, so let's have one example.

### Example: What about a Decagon (10 sides) ? Sum of Internal Angles = (n-2) × 180° = (10-2)×180° = 8×180° = 1440° And, if it is a Regular Decagon, then each internal angle = 1440°/10 = 144°