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I read in various places that CAPE is the area between temperature and the saturated adiabat through LCL from LFC to EL and approximated by $\bigl(\sum_{i=LFC}^{EL} {T_{ap}-T_e\over T_e} \overrightarrow g\bigr)\Delta z$

I have plotted the CAPE for a sounding and want to instead calculate the area from the resulting CAPE polygon. When I use the shoelace formula to do so I however always come to a number which seems too far off from the approximation to be correct. It is within the same order of magnitude but just doesn't seem right.

What is the correct way to interpret the CAPE as being this 'area' ? Can I calculate it as I did or can it only be approximated as per the formula ?

Update: changing the base from the surface reading to 1000mb brings the calculation closer to estimates however i hope someone can still comment on the accuracy of my interpretation on the meaning of 'area' and whether it really is the surface area of the CAPE polygon

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