Leibniz Formula for Pi/4 - A Pi Day Visualization/Dissection @MathVisualProofs

Uploaded March 2025 | Updated September 2026, 2 weeks ago

In this video, we explore a wonderful dissection of the unit circle (with area Pi) into regions that have areas given by differences of reciprocals of consecutive odds. When the diagram is complete, we get a visualization of the famous Madhava-Gregory-Leibniz series, or Leibniz formula, for Pi/4. We explain how this dissection comes about by investigating a family of regions bounded by two special polar curves, coming from powers of tangents of half angles.

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This animation was inspired by an article from Mitsuo Kobayashi that appeared in the April 2014 issue of Mathematics Magazine (jstor.org/stable/10.4169/math.mag.87.2.145
) see pages 145-150. That article features this dissection proof attributed to Viggo Brun.

For more Pi-related videos, check out my playlist:
youtube.com/playlist?list=PLZh9gzIvXQUsGRDvzvXc02lDIKHaVeFwq

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Leibniz Formula for Pi/4 - A Pi Day Visualization/Dissection

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