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How does the circumference-to-radius ratio of a circle change on a curved surface?

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differential geometry An annular strip (the region between two concentric circles) can be cut and bent into a helical strake that follows approximately the contour of a cylinder. Techniques of differential geometry are employed to find the dimensions of the annular strip that will best match the required curvature of the strake.

On a flat Euclidean plane, the ratio of a circle's circumference ( $C$ ) to its radius ( $r$ ) is exactly $2 π$ . However, on a curved surface, this ratio deviates from $2 π$ due to the surface's Gaussian curvature.1

The behavior of the ratio depends on the nature of the curvature:

Positive Curvature: On surfaces like a sphere, the circumference of a circle is smaller than $2 π r$ . As the radius increases, the ratio $C / r$ becomes strictly less than $2 π$ .
Negative Curvature: On surfaces with hyperbolic (saddle-like) geometry, the circumference of a circle is larger than $2 π r$ , meaning the ratio $C / r$ is greater than $2 π$ .

In differential geometry, these deviations are intrinsic properties of the surface. While a cylinder has zero Gaussian curvature and behaves like a flat plane in this regard, surfaces that cannot be flattened without distortion—such as a sphere—exhibit these characteristic changes in the relationship between circumference and radius.12