This introduction contains several questions directed to the reader. Please take your time and give thought to them. They are intended to be challenging questions, and the main purpose they serve it to make you think, not to necessarily get solved.
We all have some naive sense of 'area', but let's try to really
understand what area is. We are taught certain formulas as children,
for example, the area of a triangle is half its base times its height.
But what does this actually mean? And how do we know it is true?
In fact, the concept of area is not at all a simple one. Pythagoras, the first person to use the word "Mathematics", is associated with an ancient form of geometry called "the application of area", which is basically the attempt to truly understand what area is. In fact, it seems that there would no true way to define area, but this does not mean that we have no way of properly studying area. In order to do that, we will have to find a more humble goal than defining area -- to know when two shapes have the same area. This goal will guide us during the whole discussion presented here. Keep it in mind.
Main Question: When do two shapes have the same area?
Let's start with a simple case. Observe the following two shapes. Do they have the same area?
Since these two shapes are identical in all aspects, they must have the same area. This is a trivial example, but notice that we did not need to calculate the area of each square in order to know that they have the same area.
How about these two shapes?
These are not identical, but we still believe they have the same area. Why is that?
There are two ways to show that these two shapes have the same area -- dissection and rotation:
Let's ask a more difficult question. Do these two shapes have the same area?
This is not immediately obvious. Some people might say, just use the formula for the area of a triangle, but as stated before, how do you know it is true? In fact that formula is discovered by doing precisely what we are about to do.
By using a combination of dissection and rotation (the top-left half of the square is rotated into the triangle), we show that these two shapes have the same area.
Will this method work for all triangles and squares?
Can you find a way to show that these two shapes have the same area? And if you can't, would this mean they don't have the same area? What if you just didn't find the "trick"? In fact, how would you be able to show at all that two shapes don't have the same area?
Main Question (Modified): When do two shapes have the same area? And when do two shapes not have the same area?
We'll return to this question, but right now let's try to take a different path in order to broaden our understanding of area.