Area Problem

That's a small mathematical problem that was bugging my mind a few years ago. Back then, I had solved it by logic, but I thought I could put some numbers in order to know exactly what's going on.

The problem is that we have four shapes, and by only moving them, we get a different area.

The area of the red and yellow triangles is (21 × 8) / 2 = 84 each, and the area of the green and blue triangles is (13 × 8) + ( (13 - 8) × 13) / 2 = 136.5 each. That gives 84 × 2 + 136.5 × 2 = 441, so what's wrong with the second picture?

Please take a few moments to think about it. Sleep on it if you want! There's always that sense of pride when you success in doing something by yourself.

The answer is in the angles. They seem to be the same, but are not. The red and yellow triangle's slope is (21 / 8) = 2.625, while the green and blue triangle's slope is (13 / (13 - 8)) = 2.6. Another way to make sure is to calculate the angles like this:

That means that, on the second picture, the long edge made by the green and red triangles is not a straight line. If I exaggerate the difference in the angles (for comprehension purposes), it would give something like this, which explains the difference in areas:

A few years ago, it wasn't really this problem that I saw, but something that can be solved in exactly the same way. It looked like that:

The rest of the post is only some mathematical rambling...

If we really want to be crazy, we can verify that the empty shape inside has an area of 1 (which is the initial difference between the two areas). We should divide the problem into two large triangles, calculate the area of one triangle, then multiply by 2 to find the empty area inside. I'll keep the calculations as precise as possible.

First, we can calculate all the hypotenuses and angles:

Which results in the following triangle:

We can use trigonometry if we divide the triangle into two right triangles, and calculate the angles:

Then it's only a matter of calculating the other sides with sine and cosine:

and calculating both triangles' areas. It looks messy, but I wanted to show all the steps.

Square root of x, squared, equals x, and sin(θ) × cos(θ) = sin(2θ)/2, so that simplifies to:

We had two large triangles, so this gives us 1, which is the difference in area that was the subject of this post from the beginning.

Writing this post taught me that I should do all the equations on paper first, in case I make a mistake somewhere. Fortunately, I did not, but I was scared to have to correct all the images if I did.