Of All the Areas

In this problem we are going to be thinking about the areas of equilateral triangles.
The important thing to keep in mind is that, to make life easy for ourselves, we will measure our areas in terms of equilateral triangles too.

Here are some equilateral triangles.


If the area of the first triangle is 1, what are the areas of the others?
Do you see any patterns?
Are you surprised?
Will these patterns continue? Why?

All the triangles you have just been looking at had horizontal bases, but of course equilateral triangles can be tilted.

Here are some equilateral triangles with a tilt of 1.



Can you explain what I mean by a tilt of 1?
How do I know they are equilateral triangles? Can you convince yourself that they are?

Can you find their areas? If you need some help with this why not use the geoboard environment below and/or use the hints .

FULL SCREEN VERSION

Can you extend your ideas to find the areas of other equilateral triangles with a tilt of 1?
Is there a general rule? Can you explain it?

How about the areas of triangles with a tilt of 2, like the ones below?



What about other tilts?