Pi in other geometries

Since ancient times, dating back to at least the Rhind Papyrus some 1800 BC, it was already known that the ratio of the circumference of a circle in the plane to its diameter is a constant we know as pi. But what about pi in other geometries? Is it also a constant, and if so, does it have the same value?

First we look at the interactive applet below which measures the area CO and radius AB of a circle in Euclidean plane geometry. The blue calculation shows the area/r2 = pi. (Note: Area measurement is used here as the software unfortunately cannot measure the circumference of a circle directly.)

Drag point B to observe how the radius AB and area CO changes. What do you notice about pi?

Please enable Java for an interactive construction (with Cinderella).

Now click on the two links below to investigate pi in two other geometries.

Pi in spherical (elliptic) geometry

Pi in hyperbolic geometry

Created by Michael de Villiers with Cinderella, 15 May 2011. (Originally used as part of a talk on "Maths: Pi in the sky or bread and butter?" on International Pi Day, 14 March 2011 at Cafe Scientifique, Jive Media Africa, Pietermaritzburg.)


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