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)
© Eugene Brennan
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
© Eugene Brennan
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
© Eugene Brennan
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
© Eugene Brennan
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)
© Eugene Brennan
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
© Eugene Brennan
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.
A bicycle chainring
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"
© 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)
