Bizarre Triange Surface Area Anomaly

What in the world is going on here?

Can you explain to this **very **novice math guy why this happens?

http://kotinetti.suomi.net/jussi77/bizare.jpg

The individual pieces are exactly the same. The surface area for each component is exactly the same. The area of the composite triangle is exactly the same. Yet if the two triangles are reversed, the irregular pieces form a new empty spot. It's boggles my meager math-oriented mind.

Obviously I'm less than no math expert. My exploration was limited to drawing the shape on a piece of graph paper and calculating the area of each piece as well as the triangle as a whole. What I came to realize is that the surface area sum of the individual pieces is still somehow less than the surface area of the whole. But that's basically demonstrated by the drawing, and it only conflates the issue.

Can someone please help me understand the math at play here?

thanks to all the replies

thanks folks for indulging this non-math guy. i'm learning about the world of fibonnaci numbers, the golden ratio, and so on. it's captivating stuff for someone like me, although i know you people are much more familiar with it.

aha -- i figured that one out!

well, since ive quickly learned what to look for, i knew right away how this one works. i've also found some other interesting examples, and have moved quickly on my way to trying to believe that .999 is equal to 1. as a philosopher, it's easy enough, but if you cut a .999 repeating inch piece of wood for a 1-inch slot, no one can ever tell me it will fit perfectly....

do you folks have any other interesting example of properties of numbers or other curiosities like the one I got interested in?