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# Of All the Areas

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Age 14 to 16

Challenge Level

*Of All the Areas printable sheet*

*You may wish to print off some isometric paper or use the isometric dotty grid environment for this problem.*

*This problem follows on from Isometric Areas and More Isometric Areas.*

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?

**Can you see a relationship between the area and the length of the base of each triangle?
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"?

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

Six circular discs are packed in different-shaped boxes so that the discs touch their neighbours and the sides of the box. Can you put the boxes in order according to the areas of their bases?

ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.