Pi in spherical geometry

In the applet below, a circle is drawn on the surface of a sphere which is an example of elliptic 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 (and its faint antipode on the other side of the sphere) to see it from different positions on the surface of the sphere.

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

Explore more: Do a Google search to read up on Elliptic 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.)