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This question already has an answer here:

I want a method for find the number triangles in the under image.

enter image description here

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marked as duplicate by user147263, anomaly, Hagen von Eitzen, Mark Bennet, Vladimir Reshetnikov Oct 14 '14 at 22:31

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Start from the top. Take every juncture point and calculate the number of triangles starting, and going downward (i.e. the triangle is wholly under that point), from that point. Then add up.

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