The area of a sector at centre of a planar quadric

TL;DR

This paper derives a simple classical formula for calculating the area of a sector at the center of a planar quadric, based on boundary line lengths and angles, enhancing geometric understanding.

Contribution

It introduces a new, straightforward formula for the sector area of a planar quadric centered at the origin, based on classical geometric methods.

Findings

Formula relates sector area to boundary line lengths and angles.

Provides a closed-form expression for sector area of quadrics.

Enhances geometric analysis of planar quadrics.

Abstract

Three linearly dependent and pairwise linearly independent vectors of an euclidian space uniquely determine a planar quadric with symmetry centre in the origin. A rather simple formula for the area of an arbitrary sector at centre of such a quadric will be shown by classical methods. The formula describes that area in dependence of 1. the lengths of the two straight lines that bound the sector at two sides, 2. the length of an arbitrary straight line from the centre to the quadric arc that bounds the sector at the third side, 3. the two angles in between these three straight lines.