Measuring a Sprocket Bolt Circle with an Odd Number of Holes

On a five-hole mounting pattern, the most widely separated hole centres do not lie opposite each other across the circle. Measuring between them gives a chord shorter than the bolt-circle diameter. Treating that span as the diameter can therefore produce a wrong fitment comparison even when the measurement itself was taken carefully.

The diameter can be recovered from a known centre-to-centre chord when the holes are equally spaced on one circle. The calculation needs three pieces of information: the total hole count, the measured chord and how many hole intervals separate its endpoints. Keeping those three together is more reliable than looking for the pair that merely appears widest.

Confirm that the holes form the assumed circle

A bolt-circle model describes the centres of the mounting holes, not the outer edge of the sprocket or the circle through its tooth tips. Before calculating, establish that the relevant hole centres lie on one circle and are equally spaced. Count the complete pattern; do not count only the holes visible in a partly obstructed photograph.

The holes also need to belong to the same mounting plane and pattern. An extra locating hole, another concentric bolt circle or a non-uniform spacing arrangement can invalidate a calculation based on the apparent total. Use the applicable drawing where available and compare it with the physical part.

Round, intact holes provide a clearer basis for recovering centres than elongated, damaged or irregular openings. A formula for equal circular spacing does not turn an oval worn hole into a known centre location. If the relevant geometry is distorted or the spacing is not equal, the correct response is to obtain suitable dimensional information rather than force the measurements into the equal-spacing model.

Access must also be safe and adequate for the intended measurement. This calculation does not require an installed sprocket to be exposed, rotated or removed by an improvised method. Measurements should come from a safely accessible component or an applicable qualified inspection, with the part's identity retained.

Choose the chord and count the intervals

Number the holes around the circle in order. Let n be the number of equally spaced holes and k the number of intervals between the two centres used for the measurement. Adjacent centres have k = 1. In a five-hole pattern, going from the first hole to the third passes through two intervals, so k = 2.

The distinction is between holes and intervals. If the chosen endpoints are holes 1 and 3, three hole labels are involved in the sequence, but only two gaps separate the endpoints. Entering k = 3 by counting labels changes the geometric description. Although some symmetric cases can give the same chord, that coincidence is not a reliable way to record the measurement.

For convenience, choose the shorter route around the pattern, with k no greater than half of n. On an odd-hole pattern, the largest such value is (n − 1)/2. There is no integer k equal to n/2, which is why no pair is diametrically opposite. On an even-hole pattern, k = n/2 does exist and the corresponding chord equals the diameter.

The measured c must be the distance between the selected hole centres. An inside-edge gap or an outside-edge span has different endpoints. For two equal, round holes of diameter d, distances taken along the line joining their centres obey these geometric relationships:

centre distance = facing-edge gap + d = far-edge span − d

For example, two hypothetical 10 mm holes with a 70 mm facing-edge gap have an 80 mm centre distance. The far-edge span along that same line is 90 mm. Entering 70 or 90 mm as c would feed an edge distance into a centre-distance formula. These relationships assume the hole diameters and measurement direction are established; an arbitrary caliper span across worn openings may not meet those assumptions.

On the actual part, use a method that establishes the selected centres and retain the underlying readings and hole sizes where they were needed to derive c. Unequal, elongated or damaged holes need their own geometric assessment rather than the equal-hole shortcut.

A longer chord can be helpful when the same absolute distance-reading uncertainty applies, because it reduces the relative effect of that uncertainty on the calculated diameter. It is not automatically the best choice if access or centre location becomes less reliable. Choose the pair whose centres can be established most defensibly, then record the exact pair.

Derive the diameter from the chord

One full revolution contains 360°. For n equally spaced holes, the angle between adjacent centre radii is 360°/n. A chord spanning k intervals therefore subtends a central angle of 360°k/n.

Draw radii from the circle centre to the two hole centres. Together with the chord, they form an isosceles triangle. Bisecting that triangle halves the chord and the central angle. In either right triangle, the side opposite the half-angle is c/2 and the hypotenuse is the radius R. Therefore:

sin(180°k/n) = (c/2) / R

Since the diameter D is 2R:

D = c / sin(180°k/n)

In radians, the same expression is:

D = c / sin(kπ/n)

These are the same formula with different angle units. Use degree mode for 180°k/n or radian mode for kπ/n. Entering a degree value into a calculator set to radians can produce a plausible-looking but unrelated result.

The sine is dimensionless, so D has the same length unit as c. A chord entered in millimetres gives a diameter in millimetres. The formula does not convert inches to millimetres; any unit conversion must be handled explicitly.

A five-hole example with a second-chord check

Suppose a hypothetical five-hole pattern has an adjacent centre distance of 80.00 mm. The pattern is assumed to be circular, equally spaced and undistorted. The example is not a dimension for a stocked AFTparts sprocket.

n = 5, k = 1, c = 80.00 mm

D = 80.00 / sin(36°) ≈ 136.104 mm

For that same ideal circle, the chord from hole 1 to hole 3 spans two intervals. Its predicted length is:

c₂ = D × sin(72°) ≈ 129.443 mm

Notice that the wider 129.443 mm chord is still shorter than the 136.104 mm diameter. Calling the widest measured centre span the diameter would understate the circle size by about 6.661 mm in this example.

As an arithmetic check, divide 129.443 mm by sin(72°). The result returns approximately the same diameter, subject to rounding. On a real part, a separately established second chord also tests whether the observations are consistent with the assumed pattern. It is stronger evidence than calculating two answers from the same original chord alone.

Validate the pattern, then keep the other mounting dimensions

Repeat comparable centre distances at other positions where safe access and the method permit. Equal adjacent chords should be consistent within the measurement conditions and the actual drawing requirements. If they differ materially, investigate centre-location error, damaged holes, mixed patterns or unequal spacing before averaging them into one BCD.

The sine factor also scales measurement uncertainty. For adjacent holes on the five-hole circle, dividing by sin(36°) multiplies the chord by about 1.70; a small error in c is enlarged by the same factor in D when n and k are fixed. The three decimal places in the worked result therefore describe arithmetic, not demonstrated field accuracy. Retain the raw readings and selected pair, and report precision appropriate to the actual measurement capability.

Bolt-circle diameter is only one mounting relationship. Hole diameter, number of holes, spacing arrangement, centre bore, locating features, face offset and the applicable tolerances can still affect fit. The sprocket's track-engagement features also belong to a different part of the compatibility question. A correct bolt-circle calculation cannot establish those unmeasured properties.

When comparing an AFTparts sprocket candidate, keep the exact machine and installed part identity alongside the geometry. State whether the reported BCD came from a drawing or from a chord calculation. For the latter, include n, k, c, the units and the centre-distance method. A supplier can then check the actual geometric relationship instead of relying on an unexplained “diameter” number.

For the five-hole example, the useful comparison value is approximately 136.104 mm under the stated assumptions, accompanied by the measured 80.00 mm adjacent chord. The wider 129.443 mm span belongs in the chord record, not the diameter field. Retaining that distinction prevents the odd-hole geometry from creating a false mismatch before the rest of the mounting dimensions are even compared.

References

• ElectricScooterParts: measuring bolt-circle diameter and identifying hole-centre relationships
• ToolGrit: bolt-circle geometry and chord checks