Find the Number of Triangles/Square Type Questions

Find the Number of triangles squares and circles etc type questions are often asked in the analytical reasoning sections of different tests conducted by NTS, HEC, FPSC and other departments.

There are two main types of such puzzles:
(1) Formula type, where the answer can be found by analysis and derivation of a formula. The figure shows a large triangle with many lines. If the number of internal lines from top vertex is A, and the number of internal lines parallel to base is B, then the total number of triangles N = (A+1)(A+2)(B+1)/2
Here A=4, B=4, so N=5*6*5/2=75. If you count, you will get same answer. If A=40 and B=50, then it is very difficult to count, but the formula will easily give the answer.

(2) Observation type, where there will be no general formula, but you have to observe carefully. The figure shows a set of randomly intersecting or overlapping triangles. There is no easy formula for total number of triangles and you have to patiently and carefully count. With many triangles with more intersections or overlaps, it becomes very difficult to ensure a correct count.

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