Interior Angle

external image interior-exterior-angles.gif

Triangles

The Interior Angles of a Triangle add up to 180°

external image interior-angles-triangle1.gif
external image interior-angles-triangle2.gif

90° + 60° + 30° = 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)
external image interior-angles-square1.gif
external image interior-angles-square2.gif

90° + 90° + 90° + 90° = 360°

80° + 100° + 90° + 90° = 360°

A Square adds up to 360°


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

The Interior Angles of a Quadrilateral add 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°)

external image interior-angles-square3.gif
... and for this square they add up to 360°
... because the square can be made from two triangles!

Pentagon


external image interior-angles-pentagon.gif
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:
(Note: it is a Regular Polygon when all sides are equal, all angles are equal.)




If it is a Regular Polygon...
Shape
Sides
Sum of
Internal Angles

Shape
Each Angle
Triangle
3
180°
triangle
triangle

60°
Quadrilateral
4
360°
Quadrilateral
Quadrilateral

90°
Pentagon
5
540°
Pentagon
Pentagon

108°
Hexagon
6
720°
Hexagon
Hexagon

120°
Heptagon (or Septagon)
7
900°
external image regular-heptagon-sm.gif
128.57...°
Octagon
8
1080°
external image regular-octagon-sm.gif
135°
...
...
..
...
...
Any Polygon
n
(n-2) × 180°
external image regular-n-gon-sm.gif
(n-2) × 180° / n
That last line can be a bit hard to understand, so let's have one example:

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

external image decagon.gif

Sum of Internal Angles
= (n-2) × 180°

(10-2)×180° = 8×180°

1440°

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