How to Calculate Bolt Circle Diameter (BCD) for Chainrings and Bash Guards
Bolt circle diameter (BCD)
© Eugene Brennan
What Is a Bolt Circle Diameter (BCD) or Pitch Circle Diameter (PCD)?
As you may be 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, chainring 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 chainrings on bikes and the bash guards that cover them, and you can calculate it by measuring the distance between the mounting holes of the chainring on the spider. 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. Alternatively you can skip to the end of this guide where there are tables for various bolt spacings.
Imaginary bolt circle
© Eugene Brennan
Working Out the Formula for Bolt Circle Diameter
I've given the formula further down this guide, but the math for working it out is fairly straightforward if you want to follow it through. To do the calculations, 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) as shown in the diagram below.
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"
So:
sine θ = |opposite side| / |hypotenuse|
cosine θ = |adjacent side| / |hypotenuse|
Sine and cosine are sometimes abbreviated to sin and cos.
Sine and cosine for a right-angled triangle.
© Eugene Brennan
Angle Subtended by the Arc
Since there are 360 degrees in a full circle, then if there are N equally spaced mounting 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
Recommended
In the diagram below, there are six equally spaced holes, so:
θ = 360 / 6 = 60º
Angle subtended by arc. In this example, the angle is 360/6 = 60 degrees
© 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.
Referencing the diagram above, 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 the hole spacing A. (For more info on circles, see this guide.)
So from the diagram, we see that:
Sin (θ/2) = |opposite| / |hypotenuse| = (A/2) / R
Rearranging:
R = (A/2) / Sin (θ/2)
But D = 2R
So, D = 2 x (A/2) / Sin (θ/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)
Our final equation is:
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 give 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"
This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.
© 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)
