If the scale factor for two similar figures, in this case equilateral triangles, is 4:9 **then the ratio of the areas is the square of the scale factor; here 16:81.**

Example: If the radius of an equilateral triangle is 4, then the apothem has length 2 and the sides have length `4sqrt(3)` . Thus the area of this triangle is `1/2 a p=1/2(2)(12sqrt(3))=12sqrt(3)`

If the radius of the other triangle is 9 (thus resulting in a scale factor of 4:9) then the apothem is 4.5, and the length of the sides is `9sqrt(3)` . Thus the area of this triangle is `1/2(9/2)(27sqrt(3))=243/4 sqrt(3)` .

The ratios of their areas is `(12sqrt(3))/(243/4 sqrt(3))=48/243=16/81` or 16:81 as required.

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