A polygon is a closed flat shape with all straight sides, this definition including, for example, the triangle and the pentagon. The properties of especially the regular polygons, those with all sides the same length and all angles equal, are covered at Key Stage 2, Key Stage 3 and again at Key Stage 4. For some students this is necessary revision, for those of more marked ability it can pall dreadfully.
One extension can be to explore rolling polygons. These shapes have sides that are not straight but are arcs of circles, the radius of each arc being equal to a diameter of the figure. Perhaps the most common examples of such shapes are the 20 pence and 50 pence pieces, which are rolling heptagons. Rolling polygons have a constant diameter, so that these coins, although not circular, still always fit into a square slot for the purpose of feeding into pay machines.
There are a couple of nice web pages on the rolling triangle:
http://nrich.maths.org/mathsf/journalf/dec99/prob1.html
http://commando.me.berkeley.edu/~willchui/main.html
and just one or two mentioning rolling polygons with more sides. These shapes are sometimes called Reuleaux curves after the late 19th century engineer, Franz Reuleaux.
http://www.maa.org/mathland/mathland_10_21.html
In fact, only such curves with an odd number of sides, i.e. 3, 5, 7, .... can be constructed in this way, that is, to have their diameter always the same length whatever their orientation.
Able youngsters can be asked to find the perimeter of the 50 pence piece, without actually rolling one except to check their conclusions. This can be extended to the 20 pence piece and to a rolling triangle of known diameter. The most able might be able to hazard a rule for the general n-sided rolling polygon (n odd) and to test it for particular cases. The area of the rolling polygon of known diameter is not so easy to find precisely, but could be estimated.
For anyone interested in pursuing this, the solutions are given here.
The Rolling Heptagon
If we restrict the analysis to shapes with n pointed corners, the angle subtended by one arc at the opposite corner (2” in the sketch) can quite easily be shown to be Pi / n in radians. For the equilateral triangle this is obvious. In general, the properties of the circumscribed circle give it as half the angle subtended at the centre. Pupils will probably measure this. The radius of the sector swept out by this arc is the diameter of the shape, D. The total perimeter of the rolling polygon is therefore Pi x D.
Interestingly, this is an analogous formula, of course, to that for the circumscribed circle itself. The diameter here, however, is a different quantity, so the two perimeters are not equivalent.
The area of a rolling polygon can be made up from three components. First, the area of the inscribed regular polygon, then n extra contributions due to the curved sides. These additions can each be found as the difference between the area of the sector and that of the straight-sided triangle inside it.
The only other step is to express the total area just in terms of the number of sides, n, of the shape and its diameter. A little rearranging therefore gives:
Area An of an n-sided rolling polygon = n DČ[ sinČ(”) / tan(2”) + ” - sin(”)cos(”) ]
where ” = Pi / 2n, and n > 1 and odd.
For the case of the Reuleaux Triangle, n = 3 and ” = Pi / 6, we recover the well-known result:
A3 = DČ (Pi - 31/2 ) / 2 = 0.7048 DČ
For the case of the Rolling Heptagon (as in the case of the coins):
A7 = 0.7719 DČ
For n = 11:
A11 = 0.7800 DČ
And, of course, for very large number of sides, n (n odd):
AO = Pi (D / 2)Č = 0.7854 DČ