ABHYAS PREP: NON-VERBAL REASONING · COUNTING FIGURES

Counting triangles in a figure

Count systematically: by size, or by vertex. For lines fanning out from one vertex, any two lines and the base make a triangle: n(n − 1)/2.

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Worked examples

Example 1 · level 1

How many triangles are there in the figure?

  1. 6
  2. 15
  3. 20
  4. 21
Show solution
  1. The figure is a triangle with 5 lines from the top vertex to the base.
  2. By size (in units of the smallest triangle's area): 6 of size 1, 5 of size 2, 4 of size 3, 3 of size 4, 2 of size 5, 1 of size 6.
  3. Total: 21 triangles.

Answer: 21

Example 2 · level 2

How many triangles are there in the figure?

  1. 4
  2. 14
  3. 15
  4. 17
Show solution
  1. The figure is a triangle with 1 line from the bottom-left vertex and 2 lines from the bottom-right vertex to the opposite sides.
  2. By size (in units of the smallest triangle's area): 1 of size 1, 1 of size 1.2, 1 of size 1.8, 2 of size 3, 1 of size 3.2, 3 of size 4, 1 of size 4.8, 2 of size 6, 2 of size 8, 1 of size 12.
  3. Total: 15 triangles.

Answer: 15

Example 3 · level 3

How many triangles are there in the figure?

  1. 8
  2. 16
  3. 20
  4. 22
Show solution
  1. The figure is a square with both diagonals and the square formed by joining the midpoints of its sides.
  2. By size (in units of the smallest triangle's area): 8 of size 1, 4 of size 2, 4 of size 4, 4 of size 8.
  3. Total: 20 triangles.

Answer: 20

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