Posted 05 September 2007 - 10:17 PM

actually these fit in the category of optical illusions

In the first diagram, the red and blue triangle have different angles. W/o using trig, you can see the slope of the red triangle is portrayed as 2/5, while the blue triangle is portrayed as 3/8. In the picture with emboldened lines, the eye can't see the difference between .4 and .375, but it is there. If the angles were the same, the blue triangle would intersect halfway between the first 2 graph points, and in the second portion of that diagram, the blue shouldn't actually cover the 8 spaces spanned by the green and light blue figures. The hole comes from the fact that you cannot calculate the total area of the first triangle as (13*5)/2=32.5, because in actuality, keeping the angles to be the same, it's (12.5*5)/2=31.25. In the second picture, to keep the angles the same, you would have to add the areas because the blue wouldn't complete a triangle, so blue=7.5*3/2=11.25, green=8,lblue=7,red=2*5/2=5, total is magically 31.25

The second diagram is the same principal, the angles that are being matched between the 2 symetrical orange pieces and the 2 symetrical green pieces don't match. Again, .375 vs .4, the unseen gap is what equates to the differing apparent areas.