Unit 15 Section 2 : Angle Properties of Polygons
In this section we calculate the size of the interior and exterior angles for different regular polygons.
In a regular polygon the sides are all the same length and the interior angles are all the same size.
The following diagram shows a regular hexagon:
Note that, for any point in a polygon, the interior angle and exterior angle are on a straight line and therefore add up to 180°.
This means that we can work out the interior angle from the exterior angle and vice versa:
If you follow around the perimeter of the polygon, turning at each exterior angle, you do a complete turn of 360°.
|Interior Angle = 180° Exterior Angle||
|Exterior Angle = 180° Interior Angle|
Since the interior angles of a regular polygon are all the same size, the exterior angles must also be equal to one another.
|In every polygon, the exterior angles always add up to 360°|
To find the size of one exterior angle, we simply have to divide 360° by the number of sides in the polygon.
This also means that we can find the number of sides in a regular polygon if we know the exterior angle.
|In a regular polygon, the size of each exterior angle = 360° ÷ number of sides|
In this case, the size of the exterior angle of a regular hexagon is 60° because
360° ÷ 6 = 60° and the interior angle must be 120° because 180° 60° = 120°
We can use all the above facts to work out the answers to questions about the angles in regular polygons.
|In a regular polygon, the number of sides = 360° ÷ size of the exterior angle|
Example Question 1
A regular octagon has eight equal sides and eight equal angles.
(a) Calculate the size of each exterior angle in the regular octagon.
We do this by dividing 360° by the number of sides, which is 8.
The answer is 360° ÷ 8 = 45°.
(b) Calculate the size of each interior angle in the regular octagon.
We do this by subtracting the size of each exterior angle, which is 45°, from 180°.
The answer is 180° 45° = 135°.
Example Question 2
A regular polygon has equal exterior angles of 72°.
(a) Calculate the size of each interior angle in the regular polygon.
We do this by subtracting the exterior angle of 72° from 180°.
The answer is 180° 72° = 108°.
(b) Calculate the number of sides in the regular polygon.
We do this by dividing 360° by the size of one exterior angle, which is 72°.
The answer is 360° ÷ 72° = 5 sides.
Work out the answers to this question then click on the buttons marked
to see whether you are correct.
The interior angles of a regular polygon are all equal to 140°.
(a) What is the size of each of the exterior angles in the regular polygon?
(b) How many sides does the polygon have?
(c) What is the name of the polygon?