This comparison starts with two separate 8-inch round layers and ends with one 9×13-inch rectangle. Ignoring the source count would double the calculated factor and misrepresent the recipe.
Two rounds total 32π square inches
One 8-inch round has area 16π. Multiplying by two pans gives 32π, approximately 100.5310 square inches. The approximation is only for the decimal display; the symbolic 32π stays intact for the next step.
The rectangle has exactly 117 square inches
The 9×13 target area is exactly 9 × 13 = 117 square inches. Dividing by the source gives 117 ÷ (32π), approximately 1.163821. Pi cannot cancel because the target formula is rectangular, so the factor is correctly labelled approximate.
A 500-unit quantity becomes about 581.9103
PanRatio multiplies 500 by the unrounded internal ratio and rounds only the displayed result to ≈581.9103. This does not guarantee that a particular batter fits safely in the 9×13 pan: compare wall height, capacity, fill guidance, and recipe behavior before pouring.