Polygons Mr Red is going to walk around the outside of the polygon. The black arrow shows the way he is facing as he walks.

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Presentation transcript:

Polygons Mr Red is going to walk around the outside of the polygon. The black arrow shows the way he is facing as he walks.

What happened to the arrow as Mr Red walked around the polygon? We will repeat the walk but this time watch the arrow carefully.

Watch the arrow as Mr Red walks around the polygon

The arrow turned through This will be the case for walking around the outside of any polygon. It does not matter how many sides it has. You will always end up facing the way you started having made a whole turn of 360 0

Let us look closely at a corner The blue arrow shows the angle turned through as we pass a corner. It is outside the polygon so is called an EXTERIOR angle.

We know that if we go around all the corners of the polygon we will turn through all the exterior angles. We have also seen that we turn through This shows us that all the exterior angles together of any polygon must add up to

The sum of the exterior angles of any polygon = 360 0

What is a REGULAR polygon? All the sides are equal in length. All the angles are equal in size.

For a regular polygon All the exterior angles are the same. All the exterior angles add up to 360. To find an exterior angle we divide 360 by the number of angles (or sides since it is the same).

What is the exterior angle of a regular dodecagon? Dodecagon 12 sides so 12 exterior angles Regular so all angles the same All of them add up to 360 Share 360 equally between the 12

What is the exterior angle of a regular dodecagon? ÷ 12 = 30 0

Copy and complete this table for Regular Polygons Number of sidesNameExterior Angle 3Triangle

Copy and complete this table for Regular Polygons Number of sidesNameExterior Angle 3Triangle Square90 0 5Pentagon72 0 6Hexagon60 0 8Octagon45 0 9Nonagon Decagon36 0

Interior angle The INTERIOR angle of a polygon is the angle INSIDE the shape Exterior angle INTERIOR angle

We can see from the diagram that the exterior angle and the interior angle at any corner of a polygon together make a straight line or Exterior angle INTERIOR angle

To find the INTERIOR angle of a regular polygon First find the exterior angle by dividing 360 by the number of angles (sides). Take your answer for the exterior angle from 180 to get the interior angle.

What is the interior angle of a regular 18 sided polygon? Exterior angle is 360 ÷ 18 = 20 So each INTERIOR angle is 180 – 20= 160 0

Complete your table to show the interior angles of Regular Polygons Number of sidesNameExterior AngleInterior Angle 3Triangle Square90 0 5Pentagon72 0 6Hexagon60 0 8Octagon45 0 9Nonagon Decagon36 0

Complete your table to show the interior angles of Regular Polygons Number of sidesNameExterior AngleInterior Angle 3Triangle Square90 0 5Pentagon Hexagon Octagon Nonagon Decagon

What do the angles of an irregular polygon add up to? Here is an irregular pentagon First we put a dot inside the polygon (somewhere near the middle) Then we join every corner to the dot with a straight line First we put a dot inside the polygon (somewhere near the middle). Then we join every corner to the dot with a straight line.

We have produced 5 triangles. Each side of the polygon produces a triangle when we join its ends to the dot. 5 sides so 5 triangles Each triangle has angles that add up to So all the angles inside the shape add up to 5 x

All the angles add up to 5 x 180 = 900 But this total includes the angles around the point in the middle of the polygon. 180 They are not anything to do with the angles of the polygon so we must take them away from the 900. Angles round a point total 360. So the interior angles must add up to (5x180) – 360 = 540 0

To calculate the sum of the interior angles of an irregular polygon Multiply the number of sides by 180 (each side produces a triangle if you join it to a point in the middle). Take away 360 (the angles around the point in the middle which are not interior angles of the polygon). Answer is the sum of the interior angles of the polygon.

What do the angles of an irregular 13 sided figure add up to? 13 x 180 = – 360 = 1980 The angle sum of a 13 sided shape is

What about a polygon with n sides? What is the exterior angle of a regular polygon with n sides? 360 ÷ n What is the interior angle of a regular polygon with n sides? 180 – (360 ÷ n) What is the angle sum of an irregular n sided polygon? 180xn – 360 (180n-360)