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A photo produced with a single thin lense is provided. Only based on the photo I should calculate the diameter of the lense.

The photo shows a ruler in the foreground and a point source of light in the background. The point source is blurred and has a size of 3.5 cm on the ruler.

My problem with the question is that the diameter of a lense is not used in any common formulas because it does not change how light rays traverse the lense.

I hope to find out how the diameter of a lense alters the projected image or preferred a formula containing the diameter of a lense.

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This is a depth-of-field problem. A bit of background: photographers adjust the diameter of the lens effective diameter (inversely proportional the lens f-stop) to control the blur of the background image. Geometrically, the blur disk corresponds to the intersection of a cone drawn from the point source to the lens perimeter with the focal plane:

enter image description here

A lens cannot distinguish between a point source behind the focal plane from light emitted from the disk in the focal plane. Any light from the point source that intersects the focal plane outside this disk misses the lens. The smaller the lens (large f-stop), smaller the blur (greater the depth-of-field). If you have s2 and s3, you can figure out the lens size from the geometry.

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