Written problem using interior angles

Question

A primary school teacher gives pupils a regular black pentagon and five regular white hexagons. She tells them to glue them into a flat piece of paper in a football pattern. She says to be careful to make all of the edges and corners meet. Use interior angles to show that this cannot be done.

Answer 1

sum of interior angles of a polygon = (n-2)(180)

 

when you have a regular polygon, you can divide this sum by the # of angles to get the measure of one interior angle

 

for a regular HEXAGON:

 

sum of the interior angles = (6-2)(180) = 4(180) = 720°

 

one interior angle = 720/6 = 120°

 

When you put hexagons together, three hexagons fit perfectly because the three interior angles add to 120 + 120 + 120

 

=360° which is a full circle

 

just like 4 squares fit perfectly because each angle is 90° and 4(90) = 360

 

For a regular PENTAGON

 

sum of the interior angles = (5-2)(180) = 3(180) = 540°

one interior angle = 540/5 = 108°

 

You can't fit these together perfectly with no gaps because the number 360 is not divisible by 108

 

and you can't fit together 5 hexagons around 1 pentagon in such a way that there aren't any gaps because at any one

 

corner or vertex, their interior angles would be 120 + 120 + 108 and this doesn't complete a circle.