Because of the combined effects of gravitation and rotation the Earth's shape is roughly that of a sphere slightly flattened in the direction of its axis. For that reason in cartography the Earth is often approximated by an oblate spheroid instead of a sphere. The current World Geodetic System model in particular uses a spheroid whose radius is approximately 6378.137 km at the equator and 6356.752 km at the poles (a difference of over 21 km). Contents 1 Equation 2 Surface area 3 Volume 4 Curvature 5 See also 6 References 7 External links Equation

A spheroid centered at the "y" origin and rotated about the z axis is defined by the implicit equation

where a is the horizontal transverse radius at the equator and b is the vertical conjugate radius.1 Surface area

A prolate spheroid has surface area

where is the angular eccentricity of the prolate spheroid and e sin() is its (ordinary) eccentricity.

An oblate spheroid has surface area where is the angular eccentricity of the oblate spheroid. Volume

The volume of a spheroid (of any kind) is . If A2a is the equatorial diameter and B2b is the polar diameter the volume is . Curvature If a spheroid is parameterized as where is the reduced or parametric latitude is the longitude and and then its Gaussian curvature is and its mean curvature is Both of these curvatures are always positive so that every point on a spheroid is elliptic. See also Ellipsoid Prolate spheroid Oblate spheroid Ovoid References The computist's manual of facts and merchant's and mechanic's calculator External links Calculator: surface area of oblate spheroid Calculator: surface area of prolate spheroid