Triangles and Hexagons
In order to know where to cut slots for tabs and how long to make the cuts, intuition about shapes is more helpful than math, although that's rather stupid because intuition about shapes is part of understanding math. I try to make pop-ups as stupidly as possible, that is, with as little measuring as necessary.
A hexagon can be imagined as 6 equilateral triangles all shoved together, and imagining that can save you a lot of trouble. Then you can see two of the triangles forming a bow tie, and that bow tie can tell you exactly where and how long to make cuts for tabs, with no measuring atoll.
There are two hexagons in the photograph above, one concentric to the other. They're both placed along the main central fold, a valley fold. Imagine the larger hexagon as 6 triangles. Imagine a bow tie inside that hexagon, if it were it a propeller the lines of the triangles, as they spun horizontal with the central fold, would prove they're perfectly placed. Perfect, that is, to make slices for tabs or simply to locate the place to glue tabs.
a2 +b2 = c2 Remember that?
a2 + 1.252 =2.52
a2 + 2.5 = 6.25
a2=4.25
a = √4.25
a=2.061553
Now there's a lovely number, innit. Locating that fraction on a ruler puts you right back to estimating, or using your intuition as to exactly where to place your dot, so you may as well just skip the whole Pythagoras thing. Go on then, click for a page on how a cone is made to stand up by using two tabs attached to both sides of a fold.