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### Finding Area of a Spherical Triangle

```Date: 07/08/2004 at 20:32:13
From: Denise
Subject: spherical triangles

Assuming the earth is a sphere, and given three (latitude, longitude)
coordinates, what's the easiest way to calculate the area of the
spherical triangle formed by those three points?

For example: What is the area between coordinates 34, -105; 34.1,
-105.5; 34.2, -105.9 ?

I have found various formulas to calcuate spherical triangles, such as
Girard's theorem, but they assume I know the angles.  I don't know how
to go from coordinates to angles and take into account all the other
stuff like great circles.

I tried finding the distances between the 3 points and then using
1/4*square root (p*(p-2a)(p-2b)(p-2c)), where a,b, and c are the
lengths of the sides and p is the perimeter, but of course this
doesn't take into consideration that the Earth is not flat.

```

```
Date: 07/09/2004 at 14:07:56
From: Doctor Rick
Subject: Re: spherical triangles

Hi, Denise.

My CRC Standard Mathematical Tables contain the basic formula for the
area of a spherical triangle:

Area = pi*R^2*E/180

where

E = spherical excess of triangle, E = A + B + C - 180
A, B, C = angles of spherical triangle in degrees

This is the formula you say isn't helpful because you don't know the
angles, right? Well, the tables also have the following formula for
the spherical excess E:

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

where

a, b, c = sides of spherical triangle
s = (a + b + c)/2

You can find the sides using either the cosine formula or the
haversine formula, found on the following pages in the Dr. Math archives:

Distance using Latitude and Longitude
http://mathforum.org/library/drmath/view/54680.html

Deriving the Haversine Formula
http://mathforum.org/library/drmath/view/51879.html

Good luck!  Write back if you are still stuck and show me what you've
been able to do.

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Non-Euclidean Geometry

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