Since an apple is an irregular shape (and since we have no specifics regarding your particular apple), I don't think any stanardized formulas would apply.
One could of course use Archimedes' method for finding the volume. But I can't think of any straightforward method for determining the surface area.
You could get at least a rational approximation of surface area by simply calculating it, assuming a spherical shape after asking Archimedes about the volume. Then, measure the peelings to see how close you get.
Try cutting the apple in half and tracing it on paper; Draw axes through the trace. Locate the endpoints as well as a bunch of points on the trace.
(You will use these to find an equation). After you find the equation, you could use solids of revolution to find the volume and surface area.
You could slice it into numerous pieces and measure the radius and height of each piece.
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