I think it is impossible to measure or calculate something without using any instrument. A ruler and a calculator are minimum instruments to measure/calculate leaf area.
Joking aside, cumcumber leaf has a regular geometry. It is almost hexagonal with the center at the insertion of the petiole.
You can decompose the leaf shape in 2 arcs and 4 triangles. You can thus measure each vein from the "center" and the depths of the arc, calculcate the area of each part and sum it up.
well, you can calculate the area of one leaf of a specific type of cucumber plants, as i noticed that there are many types, then you will non-dimensionalize the area by dividing it by the square of the length L found in the photo attached then you have a general coefficient, call it IBR. then for any other leaf of the same kind (form) you can have an approximate value by only measuring the length L and square it and multiply it by IBR, so you have an approximate value of the area.
note: to calculate the first leaf area you can use the trapezoidal rule.
I would use PlotDigitzer to get the border coordinates of the leaf (you might need to place the leaf in a graph paper) and then use polyarea command in matlab to get the area.
If you want to measure leaf area, I recommend LeafByte. It's a free and open source app for iPhones and iPads. It measures leaf area and herbivory levels much more quickly than ImageJ and related software. It also automatically saves your data for you in a spreadsheet.
You can download it in the app store or read more about LeafByte at the website below.You can also check out the FAQ page for ways to measure leaves non destructively and in the field.
https://zoegp.science/leafbyte
https://zoegp.science/leafbyte-faqs
Disclaimer-I was involved in making this app, but since its free I make no money from its use.