When pi lies

Circumference equals diameter times pi for a circle, and a shaft cross-section is not a circle - it is closer to an ellipse, often a noticeably flattened one. A single calliper diameter multiplied by pi assumes the circular case, and the error it introduces is not random: it under-reads, consistently, in the direction a flattened cross-section pushes it.

Why an ellipse gives a smaller number

An ellipse has two diameters, not one - a major axis and a minor axis, at right angles to each other. A calliper closed at one orientation reads whichever of those two happens to be under the jaws, and if that is the minor axis, multiplying by pi produces a circumference smaller than the true one, because a circle with that same diameter would be the smallest ellipse consistent with the reading. What a Calliper Reads covers the instrument itself and its separate compression problem; this is the specific geometry error that survives even a perfectly gentle, perfectly placed calliper reading.

A better estimate: two diameters

Take two calliper readings at right angles - one across the widest point, one across the narrowest, ninety degrees apart - rather than one. Ramanujan's approximation for the circumference of an ellipse, using semi-axes a and b, is commonly written as pi times [3(a+b) minus the square root of (3a+b)(a+3b)], and it is accurate to a small fraction of a percent for the range of ellipses a cross-section like this falls into. The exact perimeter has no elementary formula at all: the NIST Digital Library of Mathematical Functions (section 19.9) gives it as a complete elliptic integral, and notes that a second, more elaborate Ramanujan approximation is correct within 0.023% even for an ellipse with semi-axes of 99 and 1. The practical version needs no memorised formula at the moment of measuring: take both diameters, and use an online ellipse-circumference calculator or a spreadsheet with the formula entered once. Two diameters and this formula close most of the gap a single-diameter pi calculation leaves open.

How big the single-diameter error actually is

The more flattened the ellipse, the larger the underestimate from treating it as a circle. For a mild flattening the error is small enough to ignore next to the other sources covered in Where Measurement Error Comes From; for a pronounced one it can be the largest single error in a calliper-derived girth figure, larger than the tape-stretch or tension errors that get more attention. Because the direction is always the same - a single diameter under-reads, never over-reads - it is at least a correctable bias rather than an unpredictable one, once you know to correct it.

Where the two diameters should be taken

The major and minor axes are not always obvious by eye, so take the first reading at whatever orientation looks widest, then rotate the calliper ninety degrees for the second. If a third reading at a different angle disagrees noticeably with either of the first two, the cross-section is not behaving like a simple ellipse at that point on the shaft, and the two-diameter formula is an approximation of an approximation - still better than one diameter, but worth treating as a working estimate rather than an exact figure in that case. Take both diameters at the same site along the shaft in the same session, since girth genuinely varies along the shaft's length, as the girth-varies-by-site post covers, and mixing sites between the two readings reintroduces an error the formula was never meant to absorb.

A sanity check against a tape

If a tailor's tape is available, the simplest validation of any calliper-derived figure is to compare it directly against a tape reading taken at the same site in the same session. Persistent disagreement in the same direction across several attempts points at the calliper method - either the geometry assumption or the jaw-closing force - rather than at the tape, since a tape measures the circumference directly with no shape assumption standing between the reading and the answer.

Why a tape sidesteps the whole problem

A tape or paper strip wrapped around the shaft measures the actual path around the actual shape, whatever that shape is, with no assumption about circles or ellipses at all. That is the practical reason the standard instrument guide specifies a tape for girth: it measures the quantity directly instead of inferring it from a diameter and a geometric model that may not fit. A calliper-plus-Ramanujan approach is a reasonable check on a tape reading, not a replacement for one. Once girth is corrected for cross-section shape, it can meaningfully be set against length - how length and girth relate as a ratio is one place that corrected figure gets used.

Not a different kind of claim

A corrected girth figure is still a length in centimetres with an uncertainty, not a judgement, however carefully the geometry was handled. An image scored by something like Rate Cock is answering a different question - what a picture looks like to a rating system - and AI Penis is clear that recovering an actual measurement from a photo is a separate, harder problem than any ellipse formula solves. A rating sits on a distribution of photographs rather than a corrected diameter, and a verdict on Rate Penis is an opinion rather than a geometric figure, whatever formula sits behind either instrument.