Interior Angles of Polygons
An Interior Angle is an angle inside a shape.
Triangles
The Interior Angles of a Triangle add up to 180°
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 has 4 straight sides)
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 interior 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 interior 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 interior angles add up to 540°) |
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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) | ||||
Shape | Sides | Sum of Interior Angles |
Shape | Each Angle |
---|---|---|---|---|
Triangle | 3 | 180° | 60° | |
Quadrilateral | 4 | 360° | 90° | |
Pentagon | 5 | 540° | 108° | |
Hexagon | 6 | 720° | 120° | |
Heptagon (or Septagon) | 7 | 900° | 128.57...° | |
Octagon | 8 | 1080° | 135° | |
Nonagon | 9 | 1260° | 140° | |
... | ... | .. | ... | ... |
Any Polygon | n | (n-2) × 180° | (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) ?
And it is a Regular Decagon so: Each interior angle = 1440°/10 = 144° |
Note: Interior Angles are sometimes called "Internal Angles"