Polygons A polygon is a shape made from only straight edges Polygons are named on the basis of how many sides they have You need to learn the following names: Some of the names get silly… Sides Polygon Name 12 Dodecagon 5 sides 4 sides 13 Triskaidecagon 3 sides 14 Tetrakaidecagon Triangle Quadrilateral Pentagon 19 Enneadecagon 20 Icosagon 100000 Megagon 8 sides 10100 Googolgon 6 sides 7 sides Hexagon Heptagon Octagon

Regular and irregular polygons A regular polygon is one with all sides the same and angles the same To describe all other regular polygons, just put the word ‘regular’ in front of their name Because you often use regular triangles and quadrilaterals, they have special names: Equilateral Triangle Regular pentagon Regular hexagon Square An irregular polygon has sides and angles that vary Irregular heptagon Irregular octagon

Regular and irregular polygons 1. Label each shape with its name : Regular heptagon Irregular pentagon Irregular hexagon Regular octagon 2. Explain why the following shapes are not regular polygons: a) Rhombus b) Isosceles triangle c) Rectangle ______________________________________ Sides are equal but angles aren’t ______________________________________ Only 2 sides and angles equal ______________________________________ Angles are equal but sides aren’t

Angles in Polygons The angles inside the shape at each corner are called interior angles The angle between pairs of adjacent sides are called exterior angles Can you spot the connection between interior and exterior angles? They add up to 180o at any corner Interior angle Exterior angle + In this section, we also look at rules for: the sum of the exterior angles of any polygon the sum of the interior angles of any polygon the exterior angle of a regular polygon the interior angle of a regular polygon

Sum of exterior angles What is the sum of the exterior angles of a polygon? For a polygon with n sides: the exterior angles sum to 360o

Proving that the sum of the interior angles of a triangle sum to 180o In year 7 you saw the following proof: Alternate angles Alternate angles Therefore

Sum of interior angles The interior angles of polygons can be split into triangles… Triangle Quadrilateral Pentagon Hexagon 1 triangle 2 triangles 3 triangles 4 triangles Sides Triangles Sum of interior angles Can you spot the connection between the number of sides and the sum of the interior angles? Sum of interior angles number of sides If a polygon has n sides…

Interior and exterior angles for regular polygons For any polygon with n sides: Eg regular octagon Exterior angles sum to 360o Interior angles sum to For a regular polygon: Exterior angle Interior angle Eg regular octagon Exterior angle Interior angle

Regular and irregular polygons 1. Label each shape with its name : 2. Explain why the following shapes are not regular polygons: a) Rhombus b) Isosceles triangle c) Rectangle ______________________________________ ______________________________________ ______________________________________

The sum of exterior and interior angles for any polygon 1. Work out the sum of the interior angles of an octagon 2. Work out the sum of the exterior angles of a heptagon 3. Work out the sum of the interior angles of a 12-sided polygon 4. Find the missing angles b) a)

Angles in regular polygons For any polygon with n sides: For a regular polygon: Interior angle Exterior angle + Exterior angle The exterior angles sum to 360o Interior angle The interior angles sum to 1. Use the rules to complete the following table: Polygon Sides Sum of exterior angles Sum of interior angles For regular shape Exterior angle Interior angle Pentagon Hexagon Nonagon Decagon Pendedecagon

2. Dave says “There is a regular polygon with exterior angles of 50o” 2. Dave says “There is a regular polygon with exterior angles of 50o”. Explain why Dave is wrong 3. John says “There is a regular polygon with interior angles of 170o”. Explain why John is correct 4. A pupil has three tiles. One is a regular octagon, one is a regular hexagon, and one is a square. The side length of each tile is the same. The pupil says the hexagon will fit exactly like this. Show calculations to prove that the pupil is wrong.

Problem solving with polygons 1. Find the number of sides of a regular polygon with: a) exterior angles of 15o b) interior angles of 135o c) an interior angle sum of 2700o 2. Use the rules in reverse to complete the following table: Name Sides Sum of interior angles For regular shape Exterior angle Interior angle 30 162 1080

3. The diagram shows a regular octagon. Find the size of angle x 4. Find the size of the smallest angle is this irregular pentagon