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Why does a sphere have positive Gaussian curvature?

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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.

In differential geometry, the Gaussian curvature of a surface at a given point is defined as the product of its two principal normal curvatures.12

For any point on a sphere, the surface curves in the same direction across all possible cross-sectional planes that contain the normal line. Because the principal normal curvatures are both positive and point in the same direction, their product is necessarily positive. Specifically, for a sphere of radius $r$ , both principal curvatures are equal to $\frac{1}{r}$ , resulting in a constant Gaussian curvature of $\frac{1}{r^{2}}$ .2