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calculate grid area a function of latitude and longitude on earth'ssurface

I think I found what I needed from the Dr. Math Archives. For anyone who is interested here it is:

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

Steve Knox
Colorado State University

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