Consider the black triangle

We will label the vertices of `T` by **R**,
**G**, and **B**, and
the corresponding angles of `T` by `r`, `g`, and
`b`. The letters stand for red, green, and blue, and, for
example, the vertex **R** is the vertex of `T` where
`T` is opposite a red triangle. The
angles at **R** in the black triangle `T` and in the red
triangle are opposite angles and therefore are equal. Their value
will be denoted by `r`. In fact **R** is the vertex
of two congruent lunes, one of which consists of the red triangle and
a gray triangle, and the other of which contains the black triangle
and another
red triangle. We will refer to these two lunes as the red lunes. We
will denote by `L _{r}'` the red lune which
does not contain

If you rotate the sphere you will also see a gray triangle that looks
pretty much the same as `T`. This is the
antipodal triangle `T'`. Its vertices are **R'**,
**G'**, and **B'**, which are the points antipodal to
**R**, **G**, and
**B** respectively. Since `T` and `T'` are
images of each other under the antipodal map, which is an isometry,
they have the same area.

It is important to understand the situation of each pair of like
colored lunes. Concentrate on the two blue lunes, `L _{b}`
and

To sum up, the six lunes `L _{r}`,

- The triangle
`T`is contained in each of the three lunes`L`,_{r}`L`,_{g}`L`, and in no others._{b} - The antipodal triangle
`T'`is contained in each of the three lunes`L`,_{r}'`L`,_{g}'`L`, and in no others._{b}' - Every point of the sphere which is not in
`T`or`T'`is contained in precisely one of the lunes.

Understanding the proof of Girard's Theorem comes down to
understanding the configuration of the triangle and the six lunes, and
verifying the three bulleted points. Hopefully the applets on this
page are helpful. However, by far the best way to visualize the six
lunes is by physical experimentation with an actual sphere. Get a
beach ball about 8 to 12 inches in diameter. Draw a triangle
`T` on it. Then carefully extend each side of the triangle to
a complete great circle. It will be
noticed that these great circles intersect on the
other side of the sphere and form another triangle `T'` which
is the antipodal image of `T`. Thus `T'` is
congruent to `T` and consequently has the same area.

Suppose that the three angles of `T` are **R**,
**G**, and **B**. At each of these vertices there are two lunes
of the appropriate angle that meet. One of them contains
`T`. Call this lune `L _{r}`,

Now by examining the beach ball you will be able to verify the three bulleted points.

We can sum up the bulleted points by saying that
the six lunes cover the entire sphere with the points in `T`
and `T'` covered two additional times. Therefore when we add
up the areas of the lunes we
have

Into this equation we substitute the formulas for the area of a
lune, and the surface area of a sphere of radius `R`.
Finally, using the fact
that `T` and `T'` have the same area, we get

Next, solving for the area of `T`, and collecting terms
this becomes

This last formula is called *Girard's formula*,
and the result of the formula is called *Girard's Theorem*.

We get an interesting variant if we solve for the sum of the angles:

Both formulas are interesting. The first emphasizes the area of the spherical triangle, and the second emphasizes the sum of the angles of the spherical triangle. For comparison with planar geometry, the second is especially interesting because it says precisely how much the sum of the angles of a spherical triangle exceeds two right angles, the sum of the angles for a planar triangle. That the difference involves the area of the sphere is a remarkable departure from what we would expect from our knowledge of plane geometry.

**Exercise:** Find the formula for the result of Girard's Theorem
when the angles are measured in degrees instead of radians.

The next section discusses some consequences of Girard's Theorem. | |

The previous section discusses area on the sphere. | |

Table of Contents. |

The java applets on this page were programmed by David S. Nunez.

url: http://math.rice.edu/~pcmi/sphere/gos4.html John C. Polking <polking@rice.edu> Last modified: Sun Apr 25 14:13:12 Central Daylight Time 1999