# 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?