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Want to develop a general formula to find the surface area of the highlighted area for any radius (say) r for the hemishphere Want to develop a general formula to find the surface area of the highlighted area for any radius (say) r for the hemisphere On spreading out this surface on a plane it can be measured with help of different measuring instruments, but there has to be a method to find that surface area using analytical methods.

The region is a curved surface and is only present on the hemispherical surface. You can neglect the cylindrical shape in the back.

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You don't say how your region is specified; the precise form may affect the suitability of this answer.

Radial projection away from a diameter to a circumscribed cylinder of radius $r$ is area-preserving (!) by Archimedes' theorem. If you have an analytic description of the region in cylindrical coordinates, something of the type $f(\theta) \leq z \leq g(\theta)$ for $a \leq \theta \leq b$, the area is $$ r\int_{a}^{b} \left|g(\theta) - f(\theta)\right|\, d\theta. $$

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