The Particle Analyzer calculates Area as the number of pixels forming the 8-neighbour connected particle.

The Perimeter is calculated by the getTracedPerimeter() method in the PolygonRoi.class

From the source file:

The algorithm counts edge pixels as 1 and corner pixels as sqrt(2). 
It does this by calculating the total length of the ROI boundary and 
subtracting 2-sqrt(2) for each non-adjacent corner. For example, a 1x1 
pixel ROI has a boundary length of 4 and 2 non-adjacent edges so the 
perimeter is 4-2*(2-sqrt(2)). A 2x2 pixel ROI has a boundary length 
of 8 and 4 non-adjacent edges so the perimeter is 8-4*(2-sqrt(2)).

However, there are other definitions of area and perimeter. For instance the Particles8 plugin calculates the Perimeter using the Freeman chaincode algorithm (stepping along the boundary pixels, i.e. from centre-of-pixel to centre-of-pixel and counting a distance of 1 between side-neighbours and sqrt(2) between corner-neighbours).

The Particles8 plugin calculates Area as the area enclosed by the chaincode polygon (this value is slightly smaller than the number of pixels because the polygon is traced from the centres of the boundary pixels). That plugin also returns the number of pixels in the particle as Pixels.

As a consequence of this, perimeters and areas of the same particles measured with the 2 procedures are slightly different - especially so when dealing with small particles. Also parameters that use ratios of these quantities such as Circularity are likely to be affected (since use both area and perimeter for the computation).

Which one is right? All of them are. The differences are due to different definitions of perimeter and area.

Note that there are more definitions of particle perimeter that can be used in image analysis.