BCD Calculation Formula?

[Martin109]

Is there a formula for calculated BCD (Bolt Centre Diameter?) from the distance between adjacent holes and the number of holes?

I've found some charts giving the BCD for various hole distances, but they're all 5-bolt or 4-bolt patterns, and mine's not.

[Biggsy]

From Sheldon Brown's Bicycle Glossary Bo--Bz

Multiply hole to adjacent hole distance by 1.155 for 3-bolt chainrings, or 2 for 6-bolt chainrings.

Don't say you have something other than 3, 4, 5 or 6-bolt chainrings!  

[Martin109]

From Sheldon Brown's Bicycle Glossary Bo--Bz

Multiply 1.155 for 3-bolt chainrings, 2 for 6-bolt chainrings.

Don't say you have something other than 3, 4, 5 or 6-bolt chainrings!  

Thanks, Biggsy.

Yes, they're 6-bolt TA, with a 40mm distance between the bolt hole centres.  So that means 40mm x 2 = 80mm BCD, does it?

[Biggsy]

Yeah, that'll be it.

[MSeries]

gah I'd just typed how do to it with trigonometry for any number of equally spaced holes

[Greenbank]

Imagine A and B are points at the centre of two adjacent bolt holes which you measure to be x mm apart. O is a centre of the chainring.

A full circle is 360 degress. The angle AOB is 360/n degrees where n is the number of bolt holes.

Imagine a point X half way between A and B.

OXA now makes a right angle triangle. The angle AOX is (360/n)/2 = 180/n, or pi/n in radians.
The side opposite this angle (AX) has length x/2
The hypotenuse of the triangle OXA is r, the radius of circle describing the bolt holes.

sin(theta) = opp / hypotenuse

hypoteneuse = opp / sin(theta)

r = (x/2) / sin( pi/n )

But we want the diameter (d=2r) so:-

d= x / sin( pi/n )

If we rearrange to d = x * (1/sin(pi/n)) we can compute:-

1/sin(pi/3) = 2/sqrt(3) = 1.155 (3.d.p.)
1/sin(pi/4) = sqrt(2) = 1.414 (3.d.p)
1/sin(pi/5) = 1.701 (3.d.p)
1/sin(pi/6) = 1/0.5 = 2
1/sin(pi/7) = 2.305 (3.d.p)

[Martin109]

That's way beyond my humble maths capabilities, but seems to confirm '2' as the answer I was looking for!

[tom_e]

Yes, they're 6-bolt TA, with a 40mm distance between the bolt hole centres.  So that means 40mm x 2 = 80mm BCD, does it?

Yes.  If you want to confirm it roughly, you could always just, erm, measure the diameter between two opposite bolt holes?  (since it's an even number of holes)

I realise something may be in the way, but you can probably get a ruler close enough to confirm the result pretty well?

[mark]

For any number of bolt holes, wouldn't the distance from any bolt hole to the center of the chain ring be the 1/2 the bolt circle diameter?

It's not nearly as impressive as Greenbank's method, but it seems to work about as well.

[MSeries]

For any number of bolt holes, wouldn't the distance from any bolt hole to the center of the chain ring be the 1/2 the bolt circle diameter?

It's not nearly as impressive as Greenbank's method, but it seems to work about as well.

yes but when the ring isn't fitted finding the centre of the ring is harder. And the OP asked specifically about finding the BCD given the bolt-bolt distance

[Greenbank]

For any number of bolt holes, wouldn't the distance from any bolt hole to the center of the chain ring be the 1/2 the bolt circle diameter?

If you've got an even number of bolt holes then just measure the distance between the centres of two opposite bolt holes.

If you've got an odd number of bolt holes you don't have bolt holes opposite the centre.

Measuring to the centre is all well and good in principle, but you'll be guessing where the centre is since there's nothing to mark exactly where the centre is. A small error (5mm or so) in your guess would be multiplied by 2 (now 10mm) and could lead you to buying a chainring with the wrong BCD.

[mattc]

If you're comfortable with trig, then measuring hole-to-hole is clearly the best method. But :

If you've got an odd number of bolt holes you don't have bolt holes opposite the centre.

Measuring to the centre is all well and good in principle, but you'll be guessing where the centre is since there's nothing to mark exactly where the centre is.

Is it really that hard? How about:
- measure hole-to-opposite-tooth
- measure hole-to-nearest-tooth
- subtract
? I haven't tried this, so there may be a hole in my plan ...

I would have thought that would get you close enough. (Plus you usually know where a ring has come from, so you know what the likely answer is).

[Greenbank]

Is it really that hard? How about:
- measure hole-to-opposite-tooth
- measure hole-to-nearest-tooth
- subtract
? I haven't tried this, so there may be a hole in my plan ...

What about chainrings with odd numbers of teeth?

But yes, it could be a close enough approximation although there are some BCDs that are very close to each other (neither 94 or 92 are marked as obsolete).

The trig isn't necessary if you keep the list of numbers around (like wot Sheldon has on his website). But it's the difference between giving a man a fish and teaching him to fish.

[mattc]

The trig isn't necessary if you keep the list of numbers around (like wot Sheldon has on his website). But it's the difference between giving a man a fish and teaching him to fish.

Very true - I just thought my method would only require the fisherman to be able to measure and subtract (and not have to remember any other 'stuff').
There seem to be many ways to skin a fish.

[Martin109]

Very true - I just thought my method would only require the fisherman to be able to measure and subtract (and not have to remember any other 'stuff').
There seem to be many ways to skin a fish.

All interesting, and informative.  Thanks.

[MSeries]

Is it really that hard? How about:
- measure hole-to-opposite-tooth
- measure hole-to-nearest-tooth
- subtract
? I haven't tried this, so there may be a hole in my plan ...

What about chainrings with odd numbers of teeth?

But yes, it could be a close enough approximation although there are some BCDs that are very close to each other (neither 94 or 92 are marked as obsolete).

The trig isn't necessary if you keep the list of numbers around (like wot Sheldon has on his website). But it's the difference between giving a man a fish and teaching him to fish.

when the question was asked the number of holes wasn't given so the use of trig. is the only means that is suitable for all numbers of holes. The OP asked for a formula so a table of the likely values doesn't satisfy his requirement.

The measure and subtract method doesn't take into account the bolt-bolt dimension and the OP never said a ruler was available.

[mattc]

... and the OP never said a ruler was available.

Hmmm, challenging! I'll get back to you ...