Summing and Subtracting Areas

To add the areas of two surfaces, simply measure their individual areas A1 and A2, and then add these measurements: A1 + A2. This is valid as long as the surfaces do not overlap.
example of adding two surfaces
Similarly, you can find the difference between the areas by subtracting the area of the second surface (subtrahend) from the first surface (minuend).
the difference between two surfaces

In essence, you just need to add or subtract the areas of the two surfaces.

However, the two surfaces must not overlap; they must not have any internal points in common.

If they do overlap, you cannot correctly perform the addition or subtraction of their areas.

Note: If there is an overlap, the overlapping area should be counted only once in the total sum. Simply adding both areas A1 + A2 would double-count the intersection of the two surfaces (the red area).
an example of overlapping surfaces

Additionally, it is assumed that both surfaces lie in two dimensions.

 

A Practical Example

Consider a surface with an area of A1 = 6 square meters and another surface with an area of A2 = 12 square meters.
example of two non-overlapping surfaces
The sum of their areas is 18 square meters:

$$ A_1 + A_2 = 6 \ cm^2 + 12 \ cm^2 = 18 \ cm^2 $$

The difference between their areas is 6 square meters:

$$ A_2 - A_1 = 12 \ cm^2 - 6 \ cm^2 = 6 \ cm^2 $$

It is important to note that the sum of the areas is independent of the shapes of the surfaces. The same principle applies to subtraction.

Whether the surfaces are triangles, rectangles, circles, or any other shape, what matters is the area of each surface.

Additional Observations

Here are some additional points about adding and subtracting areas:

  • The sum of two areas satisfies the commutative property of addition: $$ A_1 + A_2 = A_2 + A_1 $$
  • The sum of multiple areas satisfies the associative property of addition: $$ A_1 + (A_2 + A_3) = (A_1 + A_2) + A_3 $$

And so on.

 
 

Please feel free to point out any errors or typos, or share suggestions to improve these notes. English isn't my first language, so if you notice any mistakes, let me know, and I'll be sure to fix them.

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Surfaces (Geometry)