Bunuel wrote:

The figure above shows an equilateral triangle with three circles. Each point of contact between two circles or between a circle and the triangle is a point of tangency. If the triangle has a height of 18, what is the combined area of the two smaller circles?

A. 4π

B. 8π

C. 16π

D. 36π

E. 44π

Since there is only one way to draw an equilateral triangle with an inscribed circle, we can trust the drawing.

That is, we'll visually estimate the answer and look for the closest answer choice.

This is an Alternative approach.

The height of the triangle is 18 so its base is 18\(\sqrt{2}\) (because this is a 30-60-90 triangle) which is about 20*1.4 = 28

Then the area is about 18*28/2 = 18*14 = 140+80+32=252

Dividing our equilateral triangle into 9 identical smaller triangles, we can SEE that the two small circles are about half the area of 2 of the small triangles.

That is, about (1/2)*(2/9) of the total area or 252/9 which is a bit more than 25, say 30.

Option (B) is the only relevant choice.

** Note that with regular polygons and area-related questions it is almost always easier to estimate the answer instead of looking for an extremely complicated Precise geometrical solution.

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