Elliptic Cone Calculator

Name: Elliptic Cone - Geometry Calculator
Author: Jürgen Kummer

Calculations at a right elliptic cone. This is a right cone with an ellipse as base.
Enter the two semi axes lengths and the height and choose the number of decimal places. Then click Calculate. The lateral surface is calculated with an integral and can only be estimated here, the estimation is slightly lower than the real value.

Formulas:

L = \frac{1}{2} \cdot \int_{0}^{2 π} \sqrt{a^{2} b^{2} + h^{2} [a^{2} \sin^{2} (t) + b^{2} \cos^{2} (t)]} \cdot d t

L > \frac{1}{2} \cdot π \cdot (a \cdot \sqrt{b^{2} + h^{2}} + b \cdot \sqrt{a^{2} + h^{2}})

A = L + π \cdot a \cdot b

V = \frac{π}{3} \cdot h \cdot a \cdot b

pi:

π = 3.141592653589793...

Semi axes and height have the same unit (e.g. meter), the surfaces have this unit squared (e.g. square meter), the volume has this unit to the power of three (e.g. cubic meter). The lateral surface is the curved part of the surface area.

The error for estimating the lateral surface area is usually too low, in the range of two to six percent. Exceptions are highly elliptical bases or very flat cones, where the error can be considerably larger. For many applications, the estimate is sufficiently good, but if a very precise value is required or if the cone is very asymmetric or flat, numerical integration should be performed or the value taken from a corresponding table.

The elliptic cone is also called a conical quadric. A quadric is the solution set of a quadratic equation with several unknowns. In the case of an elliptic cone, this equation is x²/a² + y²/b² - z²/c² = 0, with the three dimensions or axes in space x, y, and z and the unknowns a, b, and c.

An elliptic cone is mirror-symmetric about two planes. These are the two planes along each of the two semiaxes of the elliptic base through the apex. It is also rotationally symmetric at angles of 180 degrees and multiples thereof. This applies if the two semiaxes are of different lengths, meaning it is not a right circular cone. Cones are not point-symmetric.

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