There were a number of correct solutions and part solutions to
this problem.
Most of you picked up the increase of
as the number of sides of the polygon increased by one,
although I am not sure that you could see "why" this was
happening. A common omission in your solutions is an attempt to
look back at the problem and try to see how the rule you have
found fits with the mathematical context itself. I hope this
makes sense.
Another small but important point is to remember that
describing
as 0.3 recurring and
as 0.6 recurring lacks elegance and makes it look as if you used
a calculator and did not think about what the mathematics was!!
Well done to:
William and Edward, Josh, Nick and Jeffy, who began to think
about the "why" of the pattern (all from King's College School,
Cambridge). Katherine and Rosie (The Mount School, York), Andrei
(School 205, Bucharest) and Clement. A partial solution was also
recieved from Vivienne (Akadmisches Gymnasium, Vienna).
The angle at the centre of the arc of each petal can be calculated as: 360° - 2 x 60° The interior angle of the polygon = 240° - 180(1-2/n)°
Giving the length of the side of the equilateral triangle as 1 unit The length of each arc (petal) can be calculated as:
As there as many arcs as there are sides of the polygon, the total length of
all the arcs is:
| n ×2p(240 - |
360
|
= p( |
n
3
|
+ 2)
|
The perimeter of the petals increases by
when the polygon increases its number of sides by one.