you have right triangles and you basically use cosine to determine the length and then you could take that length and dived the full side of the paper and from that if you get whatever you wanted great. e.g. cos(78.75degrees)… (45+22.5+11.25) is .195 very close to .2 or 1/5.

… so, budding mathematicians out there – WHY do these techniques work?

because coordinate geometry deems it so ðŸ˜€

you have right triangles and you basically use cosine to determine the length and then you could take that length and dived the full side of the paper and from that if you get whatever you wanted great. e.g. cos(78.75degrees)… (45+22.5+11.25) is .195 very close to .2 or 1/5.

Thirds:

If you plot the lines in a xy-plane, the x value is simply 1/3 (assuming that the sides of the square = 1.

Fifth:

Sorta the same thing. If you plot the lines out with the last one being at 11.25 degrees, it will just happen to go through (1/5,1)

Seventh:

I have no idea who figured this one out, but again, plot a line out to (1,1/7) and figure it out.

yeah, a student of mine that does MathsC did some of these proofs via trig also

Wow…why not just measure so u dont have extra folds?

the geometry is interesting also