# Spheroid

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A spheroid, or ellipsoid of revolution is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters.

If the ellipse is rotated about its major axis, the result is a prolate (elongated) spheroid, like a rugby ball. If the ellipse is rotated about its minor axis, the result is an oblate (flattened) spheroid, like a lentil. If the generating ellipse is a circle, the result is a sphere.

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 6,378.137 km at the equator and 6,356.752 km at the poles (a difference of over 21 km).

## Contents

### 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 $\alpha=\arccos\left(\frac{a}{b}\right)$ is the angular eccentricity of the prolate spheroid, and e = sin(α) is its (ordinary) eccentricity.

An oblate spheroid has surface area

### Volume

The volume of a spheroid (of any kind) is $\frac{4}{3}\pi a^2b \approx 4.19\, a^2b$. If A=2a is the equatorial diameter, and B=2b is the polar diameter, the volume is $\frac{1}{6}\pi A^2B \approx 0.523\, A^2B$.

### Curvature

If a spheroid is parameterized as

where $\beta\,\!$ is the reduced or parametric latitude, $\lambda\,\!$ is the longitude, and $-\frac{\pi}{2}<\beta<+\frac{\pi}{2}\,\!$ and $-\pi<\lambda<+\pi\,\!$, then its Gaussian curvature is