Obtaining Ellipsoid Dimensions from Circumferential Measurements

Abstract

The equation of a sphere can be obtained by dividing its circumference by 2π to determine its radius. The radius can then be used to calculate the sphere’s properties including its surface area and volume. However, this known circumference-radius relationship only exists for a perfect sphere, not an ellipsoid. Consequently, the dimensions, surface area, and volume of an ellipsoid can only be determined by directly measuring its axes or diameters, a process that may sometimes be difficult due to the inherent challenge in extending a measuring tool through a solid object. We therefore developed a mathematical formula for obtaining the dimensions of an ellipsoid based on its circumferential measurements. Our method leverages the fact that the three circumferences of an ellipsoid are each an ellipse with the same radii as the parent ellipsoid. A system of three equations, one equation for each ellipse, can then be used to obtain the equation of the parent ellipsoid. The mathematical formula we develop has broad application in mathematics and engineering sciences.