Sierpinski Triangle

Geometry Level 3

In the diagram, all triangles are equilateral. If $AB=16$ , then find the total area of all black triangles.

Problem and Image: Courtesy Waterloo University.

81 27 54

81\sqrt{3}

27\sqrt{3}

1 solution

Hana Wehbi
Mar 9, 2017

Notice that we have $27$ black equilateral triangles.

If $AB =16$ then the side of each black triangle is $2$ . We have $8$ black triangles on each side.

Thus, the area of each black triangle is $\sqrt{3}$ by applying the formula of the area of an equilateral triangle $\large\frac{\sqrt{3}s^2}{4}$ where $s=2$ . We have $27$ black triangles, each of area $\sqrt{3}$ which $\implies$ the desired area is $27\sqrt{3}$ .

Sierpinski Triangle

Problem and Image: Courtesy Waterloo University.

1 solution

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