The objective of this section is to
- understand how to calculate the size of interior and exterior angles for different regular polygons.
The following diagram shows a regular hexagon:
The angles marked 
are the interior angles of the hexagon.
The angles marked 
are the exterior angles of the hexagon.
In a regular polygon the sides are all the same length and the interior angles are all the same size.
Note that, for any polygon:
interior angle + exterior angle = 180°.
Since the interior angles of a regular polygon are all the same size, it follows that the exterior angles are also equal to one another.
One complete turn of the hexagon above will rotate any one exterior angle to each of the others in turn, which illustrates the following result:
The exterior angles of any polygon add up to 360°.
Calculate the sizes of the interior and the exterior angles of a regular hexagon. Hence determine the sum of the interior angles.
The exterior angles of a regular hexagon are all equal, as shown in the previous diagram.
| Therefore the exterior angle of a regular hexagon | = | |
| = | 60° | |
| So the interior angle of a regular hexagon | = | 180° − 60° |
| = | 120° |
| The sum of the interior angles | = | 6 × 120° |
| = | 720° |
The exterior angle of a regular polygon is 40° .
Calculate:
the size of the interior angle,
ShowInterior angle + exterior angle = 180°
| Interior angle | = | 180° − 40° |
| = | 140° |
the number of sides of the polygon.
ShowThe number of sides can be determined by dividing 360° by the size of the exterior angles, giving
= 9
so the polygon has 9 sides.
In a regular polygon:
exterior angle =
number of sides =