We saw previously that the time delay (and therefore the difference in the optical path lengths) for a circular loop is proportional to the area enclosed by the loop. This interesting fact actually applies to arbitrary closed loops. To prove this, we will derive the difference in arrival times of the two pulses of light for an arbitrary polygonal loop inscribed in a circle. Let the (inertial) coordinates of two consecutive mirrors separated by a ubtended angle q be

where w is the angular velocity of the device. Since light rays travel along null intervals, we have  c²(dt)² = (dx)² + (dy)²,  so the coordinate time T required for a light pulse to travel from one mirror to the next in the forward and reverse directions satisfies the equations

Typically ωT is extremely small, i.e., the polygon doesn't rotate through a very large angle in the time it takes light to go from one mirror to the next, so we can expand these equations in wT (up to second order) and collect powers of T to give the quadratic

The two roots of this polynomial are the values of T, one positive and one negative, for the co-rotating and counter-rotating solutions, so the difference in the absolute times is the sum of these roots. Hence we have

This is the net contribution of this edge to the total time increment. Recalling that the area of a regular n-sided polygon of radius R is nR²sin(2π/n)/2, the area of the triangle formed by the hub and the two mirrors is R²sinθ/2. It follows that each edge of an arbitrary polygonal loop inscribed in a circle contributes 4A_iω/(c² - v²cosθ) to the total time discrepancy, where A_i is the area of the ith triangular slice of the loop and v = ωR is the tangential speed of the mirrors. Therefore, the total discrepancy in travel times for the co-rotating and counter-rotating beams around the entire loop is simply

where A is the total area enclosed in the loop. This applies to polygons with any number of sides, including the limiting case of circular fiber-optic loops with virtually infinitely many edges (where the "mirrors" are simply the inner reflective lining of the fiber-optic cable), in which case q goes to zero and the denominator of the phase difference is simply c² - v². For realistic values of v (i.e., very small compared with c), the phase difference reduces to the well-known result  4Aw/c². It's worth noting that nothing in this derivation is unique to special relativity, because the Sagnac effect is a purely "classical" effect. The apparatus is set up as a differential device, so the relativistic effects apply equally in both directions, and hence the higher-order corrections of special relativity cancel out of the phase difference.