The surface area of an ellipsoid is the total area of the curved surface of an ellipsoid shape. It has many real-life applications, including in physics, engineering, and geodesy. In physics, it is used to calculate the surface area of planets and other celestial bodies. In engineering, it is used in designing structures and machinery with ellipsoidal shapes.

Geodesy uses the surface area of an ellipsoid to model the shape of the Earth, taking into account its equatorial bulge and flattening at the poles. Understanding the surface area of an ellipsoid is essential for accurately measuring and modeling various objects and phenomena in the world around us.

Using the symmetrical formula developed by Knud Thomsen, it is possible to keep the relative error under 1.061%.

The formula for determining the surface area of an ellipsoid is defined as: