**This post was updated on Feb. 26, 2007 to fix the links**.

Over in the comments of my “Does 64 = 65?” post, happenstance reader “ja” directs us to this graphic (link now fixed). And reader “gg” rightfully points out that:

It’s virtually the same as the first problem posed.

But, it’s **the exact same problem** since both the 64=65? and 32=33? illusions use the exact same right triangles.

The non-triangular pieces don’t matter at all in this one. Remembering that the area of any right triangle is half of base times height, take a gander:

Note that before the pieces are moved, the total area is 32. Afterward, the total area is 33. But the **real** triangle that the illusion leverages via perception (the purple fucker) has an area of 32.5. (13 x 5) / 2 = 32.5.

Something must be amiss! **Warning**: more math and trig follow. I love this shit.

Okay, we start off looking back at the 64 != 65 post. Same right triangles: One is 8 by 3, the other is 5 by 2. So we take the already-done math and bring it over:

Now for the purple fucker. It’s 13 by 5, so we can figure out everything else.

13^{2} + 5^{2} = c^{2}, so the hypotenuse is 13.9284

tan^{-1}(5/13) = 21.04°, so the other acute angle is…

90° – 21.04° = 68.96°, also…

tan^{-1}(13/5) = 68.96°

Now, as “gg” pointed out, when we put the Area 32 shapes (all yellow in the following graphic) on top of the Area 32.5 purple fucker (black in the following graphic for contrast), we can see that the purple/black fucker sticks out a bit, and the extra stickie-outie part forms a triangle.

I bet the area of that stickie-outie triangle is 0.5. Let’s go see!

Knowing all the dimensions of the stickie-outie:

… and to better visualize:

Knowing everything about the purple fucker and those red and green triangles from before, we can learn more about the stickie-outie part:

Purple 68.96° – Green 68.20° = 0.76°

Purple 21.40° – Red 20.556° = 0.484°

The other stickie-outie angle is insignificant.

Draw your perpendicular:

How long is that perp?

sin(0.76°) = perp / 5.3852, so perp is 0.07143… or,

sin(0.484°) = perp / 8.5440, so perp is 0.07217… so,

Let’s average the bitchez (due to rounding and sigfigs) to get 0.07180 for the hell of it.

Now that we know the length of the perp, we can figure out the unknown lengths of the two newly-made right triangles.

**Left-side**:

cos(0.76°) = segment / 5.3852, so left-segment length is 5.3847

**Right-side**:

cos(0.484°) = segment / 8.5440, so right-segment length is 8.5437

Add ’em up, and you get 13.9284, which was the hypotenuse of the purple fucker from before. So we’re still cool.

Now we have two right triangles to deal with, and we know everything we need to get the area of that crazy stickie-outie triangle (the difference in area between 32 and 32.5 purple fucker).

Left-side area = (5.3847 x 0.0718) / 2 = 0.1933

Right-side area = (8.5437 x 0.0718) / 2 = 0.3067

Left + Right = 0.1933 + 0.3067 = 0.5000

That solves half of the equation. Then, we can make the mystical Area 33 (black in the next graphic because it’s bigger) and lay it on top of the purple fucker (now green for contrast).

It makes an identical stickie-outie triangle.

Previous math applies.

0.5 + 0.5 = 1.0… which is why 32 seemingly equals 33… yet 32 != 33.

QED yet again, bitchez. Not that you care… still… or at all. Why do I bother?

And I thought that I was the one in the family that had the reputation for being an anal geek. Go figure. And I did read the whole damned thing. Neener-neener.