How do you find the area of irregular objects?

See below.

There are two types of irregular object shapes.

• Where the original shape can be converted in regular shapes with where measurements of each side are given.

As shown in the figure above, the irregular shape of object can be converted into possible standard regular shapes like square, rectangle, triangle, semi-circle (not in this figure) etc.

In such a case area of each sub-shape is calculated. And the sum of areas of all sub-shapes gives us the required area

• Where the original shape cannot be converted in regular shapes. In such cases there are no formulae to find the area of weird shapes like this which is drawn on a grid like the one shown in the figure below.

The resultant figure appears like the one appearing below.

Using the grid we estimate the area of the shape in terms of number of grid squares.

We count the number of grid squares those are either completely filled or more than half filled by the shape. Such squares are counted as ‘1’. If the square is less than half filled by the shape, then it is ignored. Let "Total number of '1's counted"##=N##

Often in the problem, each grid square represents a standard measurement of area – e.g., say one square metre. The result is stated as: Area of the shape is about ##Nm^2##

• These all give you a rough estimate of the area. At times, it becomes extremely important to find a area precisely, may you use a computer. Now, if you are doing it on a computer, you can employ integral calculas to find the area of an irregular shape as:

But as you go on making smaller rectangles, it takes a lot of time even for the computer, Now, Von Neumann thought of a brilliant way of doing it. Draw the shape on a wall, throw balls randomly (but uniformly distributed) to the wall. The probability that it hits the shape is given as : ##"area of the irregular shape"/"area of the wall"##

So, in code, you literally generate random points in a square that contains the shape. Then you see whether it is in the shape or not. And you continue doing this for several times (##N##). As ##N->oo##, you get the precise area of the shape.

Lets say you want to find the area of :

After few attempts:

After many attempts:

Thus, at this point,

##"count of picking the point in the area"/ N ~~ "area of the shape"/"area of the square"##

And this is very easy to do on computer.