We might not be able to square the circle but with its four sides we can treat it as a special trapezium. Who says we can’t?
Experimenting and playing around with formulae can help to show how things in mathematics are interconnected. Discovering these connections can broaden our approach to solving particular problems, when required. Many formulae we are familiar with are special cases of more general formulae and elevated to the status of formulae in their own right because they are particularly useful and encountered often. Take for example, the formula for the area of a trapezium given on page 8 of the Formula & Tables booklet approved for use in the state examinations.
Area = 1/2. (a+b) . height
This can be considered the most general case. A trapezium is a quadrilateral (four sided figure) in which there are two parallel sides. The formula given tells use that the area of the trapezium is given by the average of the two parallel sides by the perpendicular height between them. If we take the case where |a| = |b| we have the special case of a parallelogram. A square is in turn a special case of a parallelogram in which all four angles as well as all four sides are equal. If we get a little creative and make one of the parallel sides smaller and smaller until it reaches a zero length we form a triangle and get the area as 1/2 the base by the perpendicular height.
Every now and then let your mind run riot altogether, just for the fun of it, but choose your moments. While sitting examinations is not the time to do this. Examiners generally do not appreciate considering triangles or circles to have four sides as we are about to do. But by doing so we can extend the formula for the area of a trapezium to these figures. The justification is simply that it works. Others may have done this already, but I have not come across it.
So, again just for the fun of it, I am calling this Molloy’s rule until such time as I find that someone else has done it before me.
Consider the four sides of a circle as follows:
- cut a gap whose width approaches zero in the circumference
- length of circumference from one side of the gap to the other going the long way around is given by b = 2.π.r This forms one side.
- draw a radius of length r from each side of this tiny gap to the centre of the circle. This gives us two more sides, each of length r. As we did with the triangle consider the point at the centre to be the fourth side of length zero and which is parallel to the circumference and call it “a” to be consistent with the formula for the area of the trapezium used above.
- Note that the perpendicular distance from side “a” at the centre to side “b” at the circumference is the length of the radius, r, corresponding to “h” in our general formula.
Substituting into 1/2 . (a+b) . h gives the area of our four sided circle as
1/2 . (2.π .r + 0) . r = π .r2
which is correct.
There are other simple ways of deriving this result but playing around like this can not only sometimes throw up some useful or helpful ideas but because of the thinking and making of connections involved make it more likely that some result can be better learnt, remembered and recalled when required. If you work through the above flight of fancy I think you are unlikely to forget the formula for the area of a trapezium or a “four sided” circle.