MATHEMATICS AND ORIGAMI

Mathematics and Origami

7.2.2 SQUARES ́ DIFERENCE.

Let squares A and B with respective sides x, y.

Algebraically we know that:

()()x+y x−y =x^2 −y^2

The first member is equivalent to rectangle C + D

The second, to areas C + E – B

So,

C + D = C + E – B ; D = E – B

Coming back to geometry from algebra, last expression is equivalent to:

y()x−y =xy−y^2

as the latter expression is an identity, it proves that also geometrically sum times difference is

equal to the difference of squares.

7.3 AREA OF THE OTHER PARALELOGRAMS

Let ́s take the rhomboid ABCD as representative (see Point 6.5 for construction). Folding through D and
C the perpendicular to AB, we get equal triangles AED and FBC. Therefore, the area of rhomboid ABCD
is equal to that of rectangle EDCF, i.e., b×a (base times height).

7.4 AREA OF TRAPEZIUM ABCD

1- AC (valley fold).
2- EF perpendicular to AC (AC! AC).
3- BG perpendicular to EF through B (B! B; EF! EF).
4- This way CG = AB is obtained.
5- Likewise, get BH = DC.
6- Thus we get the trapezium BCGH which is equal to ABCD because it has equal angles in B and C,
as well as their three associated sides.

EA FB

a

x D b Cy

DD

xD b

C C

yC b

A

BE

C D

y B

x x

y