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 is any shape with 4 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 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:
(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° |
 |
60° |
| Quadrilateral |
4 |
360° |
 |
90° |
| Pentagon |
5 |
540° |
 |
108° |
| Hexagon |
6 |
720° |
 |
120° |
| Heptagon (or Septagon) |
7 |
900° |
 |
128.57...° |
| Octagon |
8 |
1080° |
 |
135° |
| ... |
... |
.. |
... |
... |
| 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 Regular Decagon (10 sides) ?
 |
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° |
|
Home
