Spheres and geometry

When we look at a highway, we often think of it as a straight line. Of course it is not straight—it curves with the earth. Still, if we say that the straight lines on the surface of the earth are the great circles, then we can begin to do geometry.

In this session we used a fun toy called a Lénárt Sphere, which is a sphere you can draw on with a round ruler. We used it to discuss a number of questions, including:

  • A bear leaves home, walks 200 miles south, then 200 miles west, then walks north. To her surprise, she finds that she is back home again. Where’s the bear’s house?
  • What is a straight line on the sphere? What isn’t?
  • How many “great circles” can you find between a pair of points on the sphere?
  • Suppose we call the great circles on a sphere “lines”. What are some similarities and differences between these lines and the lines we typically use in Euclidean Geometry? Some properties to think about:
    • parallel
    • perpendicular
    • number of intersections
    • shortest distance
    • lengths
  • Is it possible to have a two-sided polygon on a sphere? Why or why not?
  • What is a triangle on a sphere? Draw one.
  • Is it possible to draw a square on a sphere? Why or Why Not?
  • What shapes can be formed by three great circles on a sphere?
  • Measure the angles of triangles on a sphere. What relationship(s) can you see among the angles and the areas of the triangles?
  • The surface area of a sphere is “4 pi r squared”. Use this to find the area of
    • a region between two great circles (biangle, lune)
    • a region between three (triangle)
    • (Look for patterns in your calculations.)
  • Find a formula for the area of a triangle on a sphere by thinking about the 6 “lunes” created by three intersecting lines?

Spherical Geometry Questions Handout