explanation of area

If there are fractions of squares then they are counted as fractional square units. For example if the boundary cuts a square along its diagonal, the half on one side of the diagonal would be inside the region so we would count half a square.

This gets more complicated when the boundary is curved. One way to get a handle on this is to break each 1-unit square into four squares each 1/2 unit on a side and count the number of squares which are completely inside the region, the count again but count every square that isn't completely outside the region. We multiply each number by 1/4, since each smaller square is 1/4 of a 1-unit square, and we get two total areas. One of the areas (the one where we counted only the smaller squares completely inside the region) is less than the area of our region, the other (where we counted all the smaller squares that weren't completely outside the region) will be greater than the area of the region. at this point we don't know the area of our region, but we have it narrowed down between two numbers.

We could divide each of our smaller squares into four squares and repeat the process. This would narrow our area down even more.

We can imagine doing this again and again, each time narrowing our area down more and more.

There is some number that is always between our two areas, no matter how many times we repeat the process. That number is the area of our region.

For circles, ellipses, polygons, and regions defined by the graphs of certain well-behaved functions, we can use various techniques to find that limiting number, the area. For many figures we need to use calculus to do this. Since most people don't know calculus, we put the formulas (like pi r^2) into tables.

I imagine this is more than you wanted to know, but at least up to a point you might find it interesting.