Round a hexagon
This problem shows that the external angles of an irregular hexagon add to a circle.
Problem
Brenda is walking round an irregular hexagon (a shape with six straight sides). She starts off part of the way along one of the sides.
At each vertex (corner) she turns.
How much does she turn in total when she has walked all the way round?
Use this example to prove that the sum of the external angles of any hexagon is $360$ degrees.
(The external angle is found by extending the side and measuring the angle between the extended line and the next side.)
Getting Started
Draw a big hexagon in chalk on the palyground.
Mark a starting point along one of the sides.
Walk round and see how much you turn by the time you get back to the start.
Student Solutions
This activity proved to be a little tricky for many pupils so maybe some more thoughts need to be given about external and internal angles. What also made it harder was asking for some suggestions about proving the solution offered.
However, from Rebecca at the Baden Powell School in the UK we had this really good solution sent in.
If she went round every angle once when she went round the last angle she would be facing the way she started. She could also do this with a square or rectangle.
$90+90+90+90=360$
It also works with any triangle, because she always faces the same way when she gets back to the start.
All shapes outer angles will add up to $360 °$ because Brenda always faces the same way when she gets back to the start.
From SooyoungӬ who said they went to GCMSӬ which I assume is Gosforth Central Middle School in the U.K.Ӭ we had the following interesting suggestion:
If you shrank the hexagon as small as a dot, the exterior angle would look like a rough circle which has 360 degrees altogether.
SophiaӬ at Towcester C E Primary School send in this thought of hers:
I think the answer is 360 degrees becuse the dot goes the whole way round and ended up facing the same way as it started. I added up $90, 90, 90$ and $90$ and it added up to $360$ degrees for a square a hexagon and a triangle.
So, thank you for those and I hope that there wil be even more learning about angles and shapes for us all.
Teachers' Resources
Round a Hexagon
Brenda is walking round an irregular hexagon (a shape with six straight sides). She starts off part of the way along one of the sides.
At each vertex (corner) she turns.
How much does she turn in total when she has walked all the way round?
Use this example to prove that the sum of the external angles of any hexagon is $360$ degrees.
The external angle is found by extending the side and measuring the angle between the extended line and the next side.
Why do this problem?
This problem offers a geometric context for a generic proof. A generic proof works by using the structure of a particular example to show the result in the general case. In this case walking round any hexagon will involve turning through $360$ degrees. The external angles of the polygon are measured by extending the sides at each vertex and measured anticlockwise from the extension of the side to the adjacent side (see coloured angles below).
Possible approach
Get the children to draw hexagons in chalk on the playground. If they do this outside they can walk along the path themselves. They should start by facing along one of the lines on any of the sides. It is best to start part of the way along one side so that there is no confusion about the amount of turn involved to get to the same point facing the same way. When they get back to the starting point they will have turned through 360 degrees. It is important to turn in the same direction every time.
Key questions
Which way are you facing?
How much have you turned?
How many turns are you making to go round the hexagon?
How much have you turned when you get back to where you started?
Possible extension
It would be possible to look at walking round different polygons in the same way. Polygons with reflex angles add an interesting twist to the thinking involved.
Possible support
More support could be offered through using a robot and programming it to go around a triangle or other polygon. The children may also find it helpful to use this worksheet. to explore the external angles in one hexagon. They could also draw their own hexagons, marking in the external angles and cut them up to fit the external angles together in the same way. The problem also links to Walking round a triangle. Cutting up a hexagon with the external angles marked on and fitting them together to form $360$ degrees gives an illustration of this result.