From: Fred W. Helenius <fredh@ix.netcom.com>
Subject: Re: surface area of spherical triangle
Date: Sat, 09 Oct 1999 12:32:56 -0400
Newsgroups: sci.math

Michael Jørgensen <Michael.Joergensen@intel.com> wrote:

>Does anyone know of a (simple) proof for the formula for the surface
>area of a spherical triangle:
>    Area = R^2 * (A + B + C - Pi).

Lew Mammel posted this almost four years ago:

From: lew@ihgp3.ih.att.com (-Mammel,L.H.)
Subject: Re: What is your favorite short proof?
Date: 22 Nov 1995 00:00:00 GMT

The proof that the area of a spherical triangle is the
"spherical excess":

Extend the sides and note that the sphere is divided into
three sets of "orange slice" pairs with area 4*(A+B+C),
where A,B, and C are the angles of the triangle. These
cover the sphere, but overlap in the triangle and in its
inverse image, so the triangle's area is counted four extra
times. Hence 4*pi + 4*Area = 4*(A+B+C) and Area = A+B+C - pi.

[Example: if the spherical triangle is the first octant -- the set of
points (x,y,z) in R^3 on the unit sphere where  x>0, y>0, z>0 -- then
the 3 pairs of "orange slices" are respectively the sets of points
on the sphere where  x y > 0, where x z > 0, and where y z > 0.
For any combination of signs of  x, y, z  it is clear that exactly
one of these inequalities holds, unless either (x,y,z)  or (-x,-y,-z)
lies in the first octant. So every point is in the union of the six
"slices", but the points in our triangle and its antipode have been
counted three times, not just once. So the sum of the areas of
the six slices equals the area of the sphere plus four additional
multiples of the area of the original triangle. --djr]
==============================================================================

From: horst.kraemer@snafu.de (Horst Kraemer)
Subject: Re: Area of spherical triangle
Date: Wed, 04 Aug 1999 17:32:43 GMT
Newsgroups: sci.math
Keywords: area in terms of sides of the triangle

On Wed, 04 Aug 1999 16:41:08 +0200, "Michael Jørgensen"
<Michael.Joergensen@intel.com> wrote:

> I already know that the surface area of a spherical triangle is given by
> the formula
>     Area = A + B + C - Pi,
> where A, B, and C are the angles of the triangle. The spherical radius
> is set to one.
> 
> Is there a 'simple' expression for the Area written in terms of the
> *sides* of the triangle?

The simplest expression I know of is:


Area = 4*arctan sqrt(tan(s/2)tan((s-a)/2)tan((s-b)/2)tan((s-c)/2))

with s = (a+b+c)/2


Regards
Horst