One, Two, Three, Pi

Here’s an interesting formula:

Arctan(1) + Arctan(2) + Arctan(3) = Pi.

(Everything’s in radians, of course).

Presentation Suggestions:
Challenge students to prove this fact.

The Math Behind the Fact:
The Figure contains a hint; check that the angles of the three triangles at their common vertex add to Pi. Can you find a similar proof for the following equation?
Arctan(1/2)+Arctan(1/3) = Pi/4.

How to Cite this Page: 
Su, Francis E., et al. “One, Two, Three, Pi.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

References:
R. Nelsen, Proofs without Words. p.39.
R. Vakil, A Mathematical Mosaic, 1996. p.86.

Fun Fact suggested by:
Stan Wagon

Did you like this Fun Fact?

Click to rate it.

Average rating 3.4 / 5. Vote count: 8

No votes so far! Be the first to rate this Fun Fact

Share: