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.
Rules of length area and volume are simple for similar objects. If the ratio of lengths is a:b, then the ratio of areas is a^2:b^2 and ratios of their volumes is a^3:b^3.
Here if the ratio of lengths is 4:9
Hence ratios of their areas is equal to 4^2:9^2 or 16:81
The ratio of the lengths of 2 equilateral triangles is 4:9 therefore the ratio of their areas is 16:81.
Had these been similar tetrahedrons, then the ratios of their volumes would have been 4^3:9^3 or 64:729.
Is a circle in or about the triangles?
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