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## 'Of All the Areas' printed from http://nrich.maths.org/

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
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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?