Reference Ellipsoid and the Geodetic Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 195

Fig. 5.16 Variations in
meridian and parallel arc
lengths

Table 5.4 Variations in meridian and parallel arc lengths with latitude B (GRS80 Ellipsoid)

Length of a meridian arc (m) Length of a parallel arc (m)

1 0 1 00 1 0 1 00 B ʔ B ﾼ 1º ʔ B ﾼ ʔ B ﾼ l ﾼ 1º l ﾼ l ﾼ

0 110,574 1,842.91 30.715 111 321 1,855.36 30.923

15

110,653 1,844.15 30.736 107 552 1,792.54 29.876

30 110,861 1,847.54 30.792 96 488 1,608.13 26.802

45

111,141 1,852.20 30.870 78 848 1,341.14 21.902

60 111,421 1,856.87 30.948 55 801 930.02 15.500

75 111,623 1,860.30 31.005 28 902 481.71 8.028

90

111,694 1,861.57 31.026 0.000 0.000 0.000

Area of the Trapezoidal Map Sheet

We will next discuss the formula for calculating the area of the trapezoidal map
sheet as an application instance of the formulae for lengths of the meridian and
parallel arcs.
Topographic maps are bounded by lines of latitude and longitude, which means
that the surface of the ellipsoid is divided up into a series of map sheets according to
certain differences in longitude and latitude. As shown in Fig. 5.17 , BA and CD are
lines of longitude, and BC and AD are lines of latitude. The coordinates of point
A are (B 1 , L 1 ) and the coordinates of point C are (B 2 , L 2 ). An area element dP within
the trapezoidal map sheet (quadrilateral), and with sides rdl and MdB has:

dP ﾼ MN cos BdBdL :

b 2

Hence, with MN ﾼ

2 , obtained from ( 5.35 ) and ( 5.12 ), the area

e 2 sin 2 B

ð 1 Þ

P of the quadrilateral ABCD is: