In the applet below, a circle is drawn on the surface of the Poincare' disk which is a model of hyperbolic geometry. Again the area CO and radius AB of the circle is measured and the blue calculation shows the area/r2 = 'pi'.
Firstly, what do you notice about the value of 'pi' here? Is it the same value as in Euclidean geometry? Drag point B to observe how the radius AB and area CO changes. What do you observe about the value of 'pi' as you drag B? Drag B so the circle becomes very small or very large. What do you notice? Surprised?
Also drag A to move the circle to see it better from different positions in this geometry.
Explore more: Do a Google search to read up on Hyperbolic geometry, its history and properties.
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.