What is the rule for area?

Answer 1

There is no universal rule for area.

For some shapes there are formulae. For example, length x breadth for a rectangle (or square), pi*r2 for a circle and so on. The formulae get more complicated as the shapes become less regular.

With some shapes, even though there is no explicit formula for the shape, it is possible to divide the shape into smaller units and use formulae for each part and then sum all these parts at the end.

This could involve copying the shape onto a fine grid and counting the number of cells of the grid that are inside the shape and multiplying that total by the area of each cell.

Conceptually, this is a bit like integration. Dividing the area into smaller and ever smaller sections and counting the number that are in your area. Again, with well behave shapes, there are formulae for integration that wil make the task easy.

Eventually, though, the shapes get so irregular that there is no formula and integration is not a viable option.