How many triangles do you see in this image? (Page 2 ) | November 24, 2025

The magic formula explained simply

Looking for an elegant shortcut? For a large triangle divided into   n   segments arranged by lines parallel to the base (here   n   = 4), we can use the following formula:

T(n) = n × (n + 2) × (2n + 1) ÷ 8

Applied to   n   = 4:
T = 4 × 6 × 9 ÷ 8 = 27

This result   corresponds perfectly to our step-by-step counting. The beauty of this formula is that it “counts” all possible dimensions, including multilevel combinations. Ideal for testing your intuition… or for silencing a heated debate!

see the continuation on the next page