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Why must the radius be positive when using the area formula for a circle?

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A graph of the function f(x) = x cubed minus 3x plus 2 is shown on a blue background, with a red curve passing through key points. Local maximum and minimum points are marked with black dots at x equals negative 1 and x equals 1, illustrating important features of the curve. — function Plot of the cubic equation f(x) = x³ − 3x + 2. The plotted points are where changes in curvature occur.

In the context of mathematics, the radius $r$ in the area formula for a circle, $A = π r^{2}$ , must be a positive number because it represents a physical dimension.

While pure polynomial functions allow the independent variable to be any real number, physical constraints in geometry force variables like the radius to be positive. Because the radius defines the distance from the center to the edge of a circle, a negative value or zero would not correspond to a geometric shape in this context.1