How could you determine the area of a shape without a formula for that shape?

Answer 1

The answer depends on the complex shape. Some complex shapes can be decomposed into smaller shapes whose areas can be determined using standard formulae. It is then simply a question of adding the parts together.

For more complicated shapes, there are essentially two options: you can either use uniform laminae and mass or estimate the area using grids.

Uniform Lamina: Copy the shape onto a sheet (lamina) of material with uniform density. Cut the shape out carefully and measure its mass (or weight). Do the same for a unit square of the lamina.

Then, because the lamina is of uniform density, the ratio of the two areas is the same as the ratio of the two masses.

That is: Area of Shape/Area of Unit Square = Mass of Shape/Mass of Unit Square.

Rearranging, and noting that the area of the Unit Square is, by definition, = 1 sq unit

Area of Shape = Mass of Shape/Mass of Unit Square.

Grid Method: Copy the shape onto a grid, where each grid square has an area of G square units. Count the number of squares that are fully or mostly inside the shape. Call this number W (for whole). Count the number of squares that are approximately half inside the shape and call this number H (for half). Ignore any square that are less than half in the shape.

Then a reasonable estimate of the area of the shape is G*[W + H/2] square units. There is some arbitrariness about “mostly inside” and “approximately half” but there is no way around that. You will get more accurate results with finer grids, but they will also require much more effort in terms of counting the grid squares.