Figure 1

Remember high school geometry?
The sum of the angles of a planar triangle is always
180 degrees or Pi radians.
However, triangles
on other surfaces can behave differently!
For instance, consider a triangle on a sphere, whose
edges are "intrinsically" straight in the sense that
if you were a very tiny ant living on the sphere you
would not think the edges were bending either to the
left or right. (Such intrinsically straight lines
are called geodesics. On spheres, they correspond
to pieces of great circles whose center coincide with
the center of the sphere.)
A triangle on a sphere
has the interesting property that the sum of the angles is
greater than 180 degrees! And in fact, two triangles
with the same angles are not just similar
(as in planar geometry), they are actually congruent!
But wait, there's more: on a UNIT sphere, the AREA of
the triangle actually satisfies:
AREA of a triangle = (sum of angles)  Pi ,
where the angles are measured in radians. Cool!
Another neat fact about spherical triangles may be found
in Spherical Pythagorean Theorem.
Presentation Suggestions:
Demonstrate the assertions about angles and areas with
an example: draw a picture of a sphere and then draw
a triangle whose vertices are at the north pole
and at two distinct points on the equator.
Here's a followup question for your students:
are geodesic paths always the
shortest paths between two points?
The Math Behind the Fact:
Planar geometry is sometimes
called flat or Euclidean geometry.
The geometry on a sphere is an example of
a spherical or elliptic geometry.
Another kind of nonEuclidean geometry
is hyperbolic geometry. Spherical and hyperbolic
geometries do not satify the parallel postulate.
By the way, 3dimensional spaces can also have
strange geometries. Our universe, for instance,
seems to have a Euclidean geometry on a local scale,
but does not on a global scale. In much the same
way that a sphere is "curved", so that divergent
geodesics extending from the south pole will
meet again at the north pole,
Einstein suggested that 3space is "curved" by
the presence of matter, so that light rays (which follow
geodesics) bend near very massive objects!
How to Cite this Page:
Su, Francis E., et al. "Spherical Geometry."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
