eutrigon and co-eutrigon

in some novel and interesting work on the relationships between parts to wholes, Wayne A Roberts (Canberra, see the 'principles of nature' work) has established Pythagoras-like relationships for triangles that have one 60 degree angle

he has named these 'eutrigons' and triangles with one 120 degree angle (which he refers to as 'co-eutrigons')

he utilises unit triangles on an isometric grid
see the section on isometric areas on this blog

the relationships are straightforwardly derived from the cosine rule, substituting 60 degrees and 120 degrees but, importantly as far as appreciation is concerned, he presents justifications based on diagrams:

a 'eutrigon' is a triangle with one 60 degree angle

the eutrigon has equilateral triangles drawn on each of the sides

what is the relationship between their areas?

the triangles that look as if they are equilateral are equilateral

a dynamic version, created by Wayne Roberts:

for the co-eutrigon, this diagram is offered by Wayne Roberts:

isometric shape areas

draw several isosceles trapeziums on an isometric grid
try to find relationships between the three variables
and the number of (unit) triangles in the shape

this task is made more interesting by an involvement of three variables

it can be hard (but hopefully beneficial) for some students to grasp the notion of measuring 'area' with a unit triangle rather than unit squares

using sides that follow the isometric grid lines

the three sides can be labelled so that algebraic rules can be recorded easily

students can identify that a + b = c and give an explanation for this

rules for the 'area' usually involved two of the three variables

some other rules used the half-way line (or two lines)



it is interesting to try to relate versions of the rules for the area to each other

for an equilateral triangle

on an isometric grid, the number of unit triangles is the square of the length, as can be seen by rearranging the shape

this can also be used for isosceles trapeziums since they can be viewed as a difference of two triangles



'straight on' parallelograms (with a horizontal base and a sloping side at 60 or 120 degrees to this base) are slightly easier to analyse:

so a triangle, with one horizontal side and another going off this at 60 or 120 degrees, will have an area of: ab (half a parallelogram)

the 'area' of hexagons with rotational symmetry order 2 can build upon the area of parallelograms and/or the area of equilateral triangles

reasons for this rule can be identified from the diagrams:

three parallelograms:

or: a triangle, take away three other triangles









students can also explore the area of  'skew' equilateral triangles

describing these by isometric 'vectors' these are:
(1 , 2) top left
(3 , 1) top right
(1 , 1) bottom









for a vector (a , b) the rule for the equilateral triangle 'area' is:

and this can be justified using an equilateral triangle surrounded by three half parallelograms