Areas of Plane Figures

There are four general methods of measuring a plane area:

(a) If the geometrical figure of the boundary be known, the area can be calculated from its linear dimensions - e.g. if the boundary be a circle radius r.

Area = r²π where π = 3.14159.

The areas of composite figures consisting of triangles and circles, or parts of circles, may be determined by addition of the calculated areas of all the separate parts.

A table of areas which can be found by this method is given in Lupton's Tables, p. 7.

In case two lengths have to be measured whose product determines an area, they must both be expressed in the same unit, and their product gives the area expressed in terms of the square of that unit

(b) If the curve bounding the area can be transferred to paper divided into known small sections, eg. square millimetres, the area can be approximately determined by counting up the number of such small areas included in the bounding curve. This somewhat tedious operation is facilitated by the usual grouping of the millimetre lines in tens, every tenth line being thicker. In case the curve cuts a square millimetre in two, the amount must be estimated; but it will be generally sufficient if portions greater than a half be reckoned a whole square millimetre and less than a half zero.

(c) By transferring the curve of the boundary to a sheet of paper or metal of uniform thickness and cutting it out, and cutting out a square of the same metal of known length of side, say 2 inches, and weighing these two pieces of metal. The ratio of their weights is the ratio of the areas of the two pieces of metal The one area is known and the other may therefore be determined.

(d) By the planimeter. A pointer is made to travel round the boundary, and the area is read off directly on the graduated rim of a wheel.

For the theory of this instrument see Williamson's Integral Calculus (§149). Practical instructions are issued by the makers.

Experiment. - Draw by means of a compass a circle of 2 in. radius. Calculate or determine its area in all four ways, and compare the results.

Enter results thus:

Method	Area [sq.in.]
a	12.566
b	12.555
c	12.582
d	12.573