How to Calculate Bolt Circle Diameter (BCD) for Chainrings and Bash Guards - Owlcation - Education
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How to Calculate Bolt Circle Diameter (BCD) for Chainrings and Bash Guards

Eugene is a qualified control/instrumentation engineer Bsc (Eng) and has worked as a developer of electronics & software for SCADA systems.

Bolt circle diameter (BCD)

Bolt circle diameter (BCD)

What's a Bolt Circle Diameter (BCD) or Pitch Circle Diameter (PCD)?

As you may be aware aware if you've found this guide, a bolt circle is an imaginary circle that passes through the centre of the bolt or screw holes in a round pattern. This is typically a wheel hub on a vehicle, chain ring on a bicycle or flange on a pipe. The diameter of this circle is known as the bolt circle diameter (BCD) or pitch circle diameter (PCD). Typically BCD is quoted for bash guards covering the chain wheels on bikes and you can calculate it by measuring the distance between the fixing holes of the chain ring. If you need to order a part for a bicycle, or check the BCD of a flange and can only measure the bolt hole spacing, you need to know the formula for working it out.

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Imaginary bolt circle

Imaginary bolt circle

Working Out the Formula for Bolt Circle Diameter

To work out the formula for BCD we first need to understand what sine and cosine mean.

A right-angled triangle has one angle that measures 90 degrees. The longest side opposite this angle is called the hypotenuse. Sine and cosine are trigonometric functions of an angle and are the ratios of the lengths of the other two sides to the hypotenuse of a right-angled triangle.

Let's represent one of the angles by the Greek letter θ (theta).

Side a is called the "opposite" side and side b is called the "adjacent" side to the angle θ.

"|" around the words below means "length of"

Then:

sine θ = |opposite side| / |hypotenuse|

cosine θ = |adjacent side| / |hypotenuse|

Sine and cosine are sometimes abbreviated to sin and cos

Sine and cosine

Sine and cosine

Angle Subtended by the Arc

Since there are 360 degrees in a full circle, then if there are N equally spaced holes, the angle subtended by an arc (produced by joining the ends of the arc to the centre) on an imaginary circle passing through the bolt holes is:

θ = 360 / N

Angle subtended by arc

Angle subtended by arc

Use Sine and Cosine to Work Out the Radius

We can draw triangles and use sine and cosine to solve the triangles and find the length of the radius R, and therefore the diameter D.

Working out bolt circle diameter (BCD)

Working out bolt circle diameter (BCD)

The simplest case is when holes are equally spaced around in a circle.

If we can find the radius R of the bolt circle, then this gives us the diameter D

We calculated that:

θ = 360 / N

Also A is the spacing between holes.

D is the bolt circle diameter.

A triangle can be drawn between the two ends of an arc on the bolt circle and the centre of the circle. This can be split up into two equal right angled triangles. The length of the chord or line joining the two ends of the arc is A.

(For more info on circles see this guide :
How to Calculate Arc Length of a Circle Segment and Sector Area)

The sine of an angle in a right angled triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse.

So Sin (θ/2) = |opposite| / |hypotenuse| = (A/2) / R

Rearranging:

R = (A/2) / Sin (θ/2)

But D = 2R

So D = (A/2) / Sin (θ/2) x 2 = A/Sin (θ/2)

But we worked out θ = 360 / N

So substituting for θ gives

D = A/Sin (θ/2) = A/Sin ((360 / N)/2) = A/Sin (180 / N)


D = A/Sin (180 / N)

Equation for Bolt Circle Diameter

BCD equation. D is the BCD, N is the number of holes and A is the hole spacing

BCD equation. D is the BCD, N is the number of holes and A is the hole spacing

Working Out the BCD for a Chainring or Bash Guard

All you need to do to calculate the BCD is measure the hole spacing A between adjacent holes, count the number of holes N and plug them into the equation above to calculate the BCD.

Tables of Bolt Circle Diameters

The tables below gives the BCD for hole spacings from 10 to 200 mm, 1 to 12 inches and 3 to 8 holes.

Table 1 - Bolt Circle Diameters for Hole Spacings From 10 to 200 mm

   Number of Holes   

Hole Spacing A (mm)

3

4

5

6

7

8

10

11.5

14.1

17.0

20

23.0

26.1

11

12.7

15.6

18.7

22

25.4

28.7

12

13.9

17.0

20.4

24

27.7

31.4

13

15.0

18.4

22.1

26

30.0

34.0

14

16.2

19.8

23.8

28

32.3

36.6

15

17.3

21.2

25.5

30

34.6

39.2

16

18.5

22.6

27.2

32

36.9

41.8

17

19.6

24.0

28.9

34

39.2

44.4

18

20.8

25.5

30.6

36

41.5

47.0

19

21.9

26.9

32.3

38

43.8

49.6

20

23.1

28.3

34.0

40

46.1

52.3

21

24.2

29.7

35.7

42

48.4

54.9

22

25.4

31.1

37.4

44

50.7

57.5

23

26.6

32.5

39.1

46

53.0

60.1

24

27.7

33.9

40.8

48

55.3

62.7

25

28.9

35.4

42.5

50

57.6

65.3

26

30.0

36.8

44.2

52

59.9

67.9

27

31.2

38.2

45.9

54

62.2

70.6

28

32.3

39.6

47.6

56

64.5

73.2

29

33.5

41.0

49.3

58

66.8

75.8

30

34.6

42.4

51.0

60

69.1

78.4

31

35.8

43.8

52.7

62

71.4

81.0

32

37.0

45.3

54.4

64

73.8

83.6

33

38.1

46.7

56.1

66

76.1

86.2

34

39.3

48.1

57.8

68

78.4

88.8

35

40.4

49.5

59.5

70

80.7

91.5

36

41.6

50.9

61.2

72

83.0

94.1

37

42.7

52.3

62.9

74

85.3

96.7

38

43.9

53.7

64.6

76

87.6

99.3

39

45.0

55.2

66.4

78

89.9

101.9

40

46.2

56.6

68.1

80

92.2

104.5

41

47.3

58.0

69.8

82

94.5

107.1

42

48.5

59.4

71.5

84

96.8

109.8

43

49.7

60.8

73.2

86

99.1

112.4

44

50.8

62.2

74.9

88

101.4

115.0

45

52.0

63.6

76.6

90

103.7

117.6

46

53.1

65.1

78.3

92

106.0

120.2

47

54.3

66.5

80.0

94

108.3

122.8

48

55.4

67.9

81.7

96

110.6

125.4

49

56.6

69.3

83.4

98

112.9

128.0

50

57.7

70.7

85.1

100

115.2

130.7

51

58.9

72.1

86.8

102

117.5

133.3

52

60.0

73.5

88.5

104

119.8

135.9

53

61.2

75.0

90.2

106

122.2

138.5

54

62.4

76.4

91.9

108

124.5

141.1

55

63.5

77.8

93.6

110

126.8

143.7

56

64.7

79.2

95.3

112

129.1

146.3

57

65.8

80.6

97.0

114

131.4

148.9

58

67.0

82.0

98.7

116

133.7

151.6

59

68.1

83.4

100.4

118

136.0

154.2

60

69.3

84.9

102.1

120

138.3

156.8

61

70.4

86.3

103.8

122

140.6

159.4

62

71.6

87.7

105.5

124

142.9

162.0

63

72.7

89.1

107.2

126

145.2

164.6

64

73.9

90.5

108.9

128

147.5

167.2

65

75.1

91.9

110.6

130

149.8

169.9

66

76.2

93.3

112.3

132

152.1

172.5

67

77.4

94.8

114.0

134

154.4

175.1

68

78.5

96.2

115.7

136

156.7

177.7

69

79.7

97.6

117.4

138

159.0

180.3

70

80.8

99.0

119.1

140

161.3

182.9

71

82.0

100.4

120.8

142

163.6

185.5

72

83.1

101.8

122.5

144

165.9

188.1

73

84.3

103.2

124.2

146

168.2

190.8

74

85.4

104.7

125.9

148

170.6

193.4

75

86.6

106.1

127.6

150

172.9

196.0

76

87.8

107.5

129.3

152

175.2

198.6

77

88.9

108.9

131.0

154

177.5

201.2

78

90.1

110.3

132.7

156

179.8

203.8

79

91.2

111.7

134.4

158

182.1

206.4

80

92.4

113.1

136.1

160

184.4

209.1

81

93.5

114.6

137.8

162

186.7

211.7

82

94.7

116.0

139.5

164

189.0

214.3

83

95.8

117.4

141.2

166

191.3

216.9

84

97.0

118.8

142.9

168

193.6

219.5

85

98.1

120.2

144.6

170

195.9

222.1

86

99.3

121.6

146.3

172

198.2

224.7

87

100.5

123.0

148.0

174

200.5

227.3

88

101.6

124.5

149.7

176

202.8

230.0

89

102.8

125.9

151.4

178

205.1

232.6

90

103.9

127.3

153.1

180

207.4

235.2

91

105.1

128.7

154.8

182

209.7

237.8

92

106.2

130.1

156.5

184

212.0

240.4

93

107.4

131.5

158.2

186

214.3

243.0

94

108.5

132.9

159.9

188

216.6

245.6

95

109.7

134.4

161.6

190

219.0

248.2

96

110.9

135.8

163.3

192

221.3

250.9

97

112.0

137.2

165.0

194

223.6

253.5

98

113.2

138.6

166.7

196

225.9

256.1

99

114.3

140.0

168.4

198

228.2

258.7

100

115.5

141.4

170.1

200

230.5

261.3

101

116.6

142.8

171.8

202

232.8

263.9

102

117.8

144.2

173.5

204

235.1

266.5

103

118.9

145.7

175.2

206

237.4

269.2

104

120.1

147.1

176.9

208

239.7

271.8

105

121.2

148.5

178.6

210

242.0

274.4

106

122.4

149.9

180.3

212

244.3

277.0

107

123.6

151.3

182.0

214

246.6

279.6

108

124.7

152.7

183.7

216

248.9

282.2

109

125.9

154.1

185.4

218

251.2

284.8

110

127.0

155.6

187.1

220

253.5

287.4

111

128.2

157.0

188.8

222

255.8

290.1

112

129.3

158.4

190.5

224

258.1

292.7

113

130.5

159.8

192.2

226

260.4

295.3

114

131.6

161.2

193.9

228

262.7

297.9

115

132.8

162.6

195.6

230

265.0

300.5

116

133.9

164.0

197.4

232

267.4

303.1

117

135.1

165.5

199.1

234

269.7

305.7

118

136.3

166.9

200.8

236

272.0

308.3

119

137.4

168.3

202.5

238

274.3

311.0

120

138.6

169.7

204.2

240

276.6

313.6

121

139.7

171.1

205.9

242

278.9

316.2

122

140.9

172.5

207.6

244

281.2

318.8

123

142.0

173.9

209.3

246

283.5

321.4

124

143.2

175.4

211.0

248

285.8

324.0

125

144.3

176.8

212.7

250

288.1

326.6

126

145.5

178.2

214.4

252

290.4

329.3

127

146.6

179.6

216.1

254

292.7

331.9

128

147.8

181.0

217.8

256

295.0

334.5

129

149.0

182.4

219.5

258

297.3

337.1

130

150.1

183.8

221.2

260

299.6

339.7

131

151.3

185.3

222.9

262

301.9

342.3

132

152.4

186.7

224.6

264

304.2

344.9

133

153.6

188.1

226.3

266

306.5

347.5

134

154.7

189.5

228.0

268

308.8

350.2

135

155.9

190.9

229.7

270

311.1

352.8

136

157.0

192.3

231.4

272

313.4

355.4

137

158.2

193.7

233.1

274

315.8

358.0

138

159.3

195.2

234.8

276

318.1

360.6

139

160.5

196.6

236.5

278

320.4

363.2

140

161.7

198.0

238.2

280

322.7

365.8

141

162.8

199.4

239.9

282

325.0

368.5

142

164.0

200.8

241.6

284

327.3

371.1

143

165.1

202.2

243.3

286

329.6

373.7

144

166.3

203.6

245.0

288

331.9

376.3

145

167.4

205.1

246.7

290

334.2

378.9

146

168.6

206.5

248.4

292

336.5

381.5

147

169.7

207.9

250.1

294

338.8

384.1

148

170.9

209.3

251.8

296

341.1

386.7

149

172.1

210.7

253.5

298

343.4

389.4

150

173.2

212.1

255.2

300

345.7

392.0

151

174.4

213.5

256.9

302

348.0

394.6

152

175.5

215.0

258.6

304

350.3

397.2

153

176.7

216.4

260.3

306

352.6

399.8

154

177.8

217.8

262.0

308

354.9

402.4

155

179.0

219.2

263.7

310

357.2

405.0

156

180.1

220.6

265.4

312

359.5

407.6

157

181.3

222.0

267.1

314

361.8

410.3

158

182.4

223.4

268.8

316

364.2

412.9

159

183.6

224.9

270.5

318

366.5

415.5

160

184.8

226.3

272.2

320

368.8

418.1

161

185.9

227.7

273.9

322

371.1

420.7

162

187.1

229.1

275.6

324

373.4

423.3

163

188.2

230.5

277.3

326

375.7

425.9

164

189.4

231.9

279.0

328

378.0

428.6

165

190.5

233.3

280.7

330

380.3

431.2

166

191.7

234.8

282.4

332

382.6

433.8

167

192.8

236.2

284.1

334

384.9

436.4

168

194.0

237.6

285.8

336

387.2

439.0

169

195.1

239.0

287.5

338

389.5

441.6

170

196.3

240.4

289.2

340

391.8

444.2

171

197.5

241.8

290.9

342

394.1

446.8

172

198.6

243.2

292.6

344

396.4

449.5

173

199.8

244.7

294.3

346

398.7

452.1

174

200.9

246.1

296.0

348

401.0

454.7

175

202.1

247.5

297.7

350

403.3

457.3

176

203.2

248.9

299.4

352

405.6

459.9

177

204.4

250.3

301.1

354

407.9

462.5

178

205.5

251.7

302.8

356

410.2

465.1

179

206.7

253.1

304.5

358

412.6

467.7

180

207.8

254.6

306.2

360

414.9

470.4

181

209.0

256.0

307.9

362

417.2

473.0

182

210.2

257.4

309.6

364

419.5

475.6

183

211.3

258.8

311.3

366

421.8

478.2

184

212.5

260.2

313.0

368

424.1

480.8

185

213.6

261.6

314.7

370

426.4

483.4

186

214.8

263.0

316.4

372

428.7

486.0

187

215.9

264.5

318.1

374

431.0

488.7

188

217.1

265.9

319.8

376

433.3

491.3

189

218.2

267.3

321.5

378

435.6

493.9

190

219.4

268.7

323.2

380

437.9

496.5

191

220.5

270.1

324.9

382

440.2

499.1

192

221.7

271.5

326.6

384

442.5

501.7

193

222.9

272.9

328.4

386

444.8

504.3

194

224.0

274.4

330.1

388

447.1

506.9

195

225.2

275.8

331.8

390

449.4

509.6

196

226.3

277.2

333.5

392

451.7

512.2

197

227.5

278.6

335.2

394

454.0

514.8

198

228.6

280.0

336.9

396

456.3

517.4

199

229.8

281.4

338.6

398

458.6

520.0

200

230.9

282.8

340.3

400

461.0

522.6

Table 2 - Bolt Circle Diameters for Hole Spacings From 1 to 12 Inches

Bolt circle diameter. Value in brackets is to nearest 1/16"

Bolt circle diameter. Value in brackets is to nearest 1/16"

© 2018 Eugene Brennan

Comments

신광철 on March 18, 2020:

Good information. However, I would like to suggest to deduct bolt hole diameter. Then we can simply measure the distance of the each hole's quadrant point.

A

D= ---------------- - Øh

sin(180/n)