Area of a Latitude-Longitude Rectangle

Date: 07/31/2003 at 16:36:02
From: Stan
Subject: Area of Section on Earth's Surface for "Spherical Rectangle"

Given the latitudes and longitudes of four points on the Earth's 
surface, how do you calculate the surface area enclosed by the four 
points?  

The four points are chosen so that they lie on only two lines of 
latitude and two lines of longitude. For example, suppose the points 
are (15N,25E), (15N,40E), (60N,25E), and (60N,40E).  The enclosed area 
is defined by the polygon constructed using the four segments of 
latitude and longitude lines connecting adjacent points. 

The area calculation is complicated by the fact that the latitude and 
longitude line segments are actually curves on the Earth's surface 
and that the distance between points separated by X degrees of 
longitude varies according to the cosine of the latitude.

I assume that one needs to first convert the point locations to 
spherical coordinates and then compute the area.

Date: 07/31/2003 at 21:11:50
From: Doctor Rick
Subject: Re: Area of Section on Earth's Surface for "Spherical 
Rectangle"

Hi, Stan.

What you describe is not really a spherical polygon. That term is 
reserved for figures made of arcs of great circles. Lines of longitude 
are great circles, but lines of latitude are not (except for the 
equator). Therefore your figure is not a spherical polygon.

Actually, if a rectangle is defined as a quadrilateral with four right 
angles, there is no such thing as a spherical rectangle. The sum of 
the angles of a spherical quadrilateral is GREATER than 360 degrees.

The following page from the Dr. Math Archives describes how to find 
the area of a true spherical rectangle in the sense of a spherical 
quadrilateral with four EQUAL angles (greater than 90 degrees). It is 
NOT what you want (if you really know what you want and you have 
described it accurately).

   Spherical 'Rectangles'
   http://mathforum.org/library/drmath/view/60748.html 

Now we can lay aside the question of terminology and consider your 
figure, whatever we call it. I'll call it a lat-long rectangle. I 
helped a student with the same problem some time ago. We started with 
the formula for the area of the earth between a line of latitude and 
the north pole (the area of a spherical cap, listed in the Dr. Math 
FAQ on Geometric Formulas).

  A = 2*pi*R*h

where R is the radius of the earth and h is the perpendicular distance 
from the plane containing the line of latitude to the pole. We can 
calculate h using trigonometry as

  h = R*(1-sin(lat))

Thus the area north of a line of latitude is

  A = 2*pi*R^2(1-sin(lat))

The area between two lines of latitude is the difference between the 
area north of one latitude and the area north of the other latitude:

  A = |2*pi*R^2(1-sin(lat2)) - 2*pi*R^2(1-sin(lat1))|
    = 2*pi*R^2 |sin(lat1) - sin(lat2)|

The area of a lat-long rectangle is proportional to the difference in 
the longitudes. The area I just calculated is the area between 
longitude lines differing by 360 degrees. Therefore the area we seek 
is

  A = 2*pi*R^2 |sin(lat1)-sin(lat2)| |lon1-lon2|/360
    = (pi/180)R^2 |sin(lat1)-sin(lat2)| |lon1-lon2|

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/