Challenge Level

*This problem follows on from Isometric Areas and More Isometric Areas.
You may wish to print off some isometric paper or use the isometric dotty grid environment for this problem.*

When working on an isometric grid, we can measure areas in terms of equilateral triangles instead of squares.

Here are some equilateral triangles.

If the area of the smallest triangle is 1 unit, what are the areas of the other triangles?

Will the pattern continue?

Can you explain why?

All the triangles in the first image had horizontal bases, but it is also possible to draw "tilted" equilateral triangles.

These triangles all have a "tilt" of 1.

Can you explain why your rule works?

What about areas of triangles with other "tilts"?