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Bolt patterns using basic trigonometry
- Thread starter Martin W
- Start date
- Joined
- Feb 1, 2015
- Messages
- 9,981
He makes that complicated. For bolt circles for a hole pattern, I measure diameter of the hole first. If I have a digital caliper, I zero the caliper at that diameter. If not, I record the diameter. Then I measure the inside to inside distance between any two holes. On a digital caliper, this will read the center to center distance directly. On a dial caliper, the hole diameter has to be added to the measurement.
If measuring the bolt circle for a pattern of studs, I zero the caliper on the stud diameter and measure outside to outside distance to directly read the center to center distance. With a dial caliper, the stud diameter is subtracted . from the measurement.
I preference is to measure the distance between two holes that are furthest apart for greatest accuracy but the distance between adjacent holes can be done as well.
For bolt pattern with an even number of holes, the diameter of the bolt circle can be measured directly. For an odd number of holes or for circles too large to measure the diameter. accurately, some math will have to be done. If one doesn't have a dedicated scientific calculator, most all smart phones have one built in. If you open the calculator app, there is an option for choosing a scientific calculator rather than a simple calculator.
The first step is to make a sketch of the bolt pattern. For all patterns with equally spaced holes, the angle between adjacent holes is 360/n where n is the number of holes. As an example, here is a drawing of a 7 hole bolt pattern on a 5" bolt circle. The angle between adjacent holes is 360/7 = 51.43º. Draw a line to represent your measurement and write in the center to center distance. For adjacent holes in the example, the distance is 2.1694" Next, draw a line from the center of the bolt circle to the midpoint if the previous line and write in half the measured distance (1.0847"). Two new equal right triangles are created by this line, each with half the total angle or 25.71º. The hypotenuse of the triangles is equal to the radius of the bolt circle and the side opposite the the 25.71º angle is half the measured distance. The side opposite the angle divided by the hypotenuse is equal to the sine of the angle. Rearranging, the radius of the bolt circle is equal to half the measured distance divided by the sine of the angle. If the measured distance between holes is a, the angle between holes is A, and the radius of the bolt circle is r, the bolt circle diameter, d, is d = 2*r = 2*a/2*sin(A/2) = a/sin(A/2) = a/sin(360/2*n) =a/sin(180/n)
This method can be generalized to measuring between any two holes. Again referring to the drawing, measuring the distance between holes that are three holes apart. The angle between the holes is 3 times the angle between adjacent holes or 154.29º and the half angle is 77. 14º. The measured distance is 4.8746" and half that is 2.4373". The same trigonometric relationships as in the previous example still hold. If m is the number of holes between the measured holes, the total angle is (m+1)*360/n and the half angle is (m+1)*180/n. The diameter now becomes
d = a/sin((m+1)*180/n).
This seems rather complicated but in reality, it is but a few key strokes on a scientific calculator. Using the latter example. enter 180 * 3/7 = 77.1428. Take the sine of that angle = .97493, invert it to get 1.02572, and multiply it by the measured center to center distance, 4.8746" to get 4.99996" for the bolt circle.
Now for coordinates for the holes. This is actually quite simple and straightforward.
Determine the angle between adjacent holes bu dividing 360 by the number of holes in the bolt circle. write that angle down. With the first hole at top (x=0, y =1), the next hole going clockwise is rotated by the angle between adjacent holes and the following hole rotated by that same amount or twice the angle from the top hole and so on for all holes where the angle is less than or equal to 180º. There is no need to go any further because the remaining holes are a mirror image of these. write the angles for the holes down. For each of those holes, take the sine of the angle. Multiply those values by the radius of the bolt circle. These are the x ordinates. Then take the cosine of those angles. Multiply those values by the radius of the bolt circle. These are the y ordinates.
If measuring the bolt circle for a pattern of studs, I zero the caliper on the stud diameter and measure outside to outside distance to directly read the center to center distance. With a dial caliper, the stud diameter is subtracted . from the measurement.
I preference is to measure the distance between two holes that are furthest apart for greatest accuracy but the distance between adjacent holes can be done as well.
For bolt pattern with an even number of holes, the diameter of the bolt circle can be measured directly. For an odd number of holes or for circles too large to measure the diameter. accurately, some math will have to be done. If one doesn't have a dedicated scientific calculator, most all smart phones have one built in. If you open the calculator app, there is an option for choosing a scientific calculator rather than a simple calculator.
The first step is to make a sketch of the bolt pattern. For all patterns with equally spaced holes, the angle between adjacent holes is 360/n where n is the number of holes. As an example, here is a drawing of a 7 hole bolt pattern on a 5" bolt circle. The angle between adjacent holes is 360/7 = 51.43º. Draw a line to represent your measurement and write in the center to center distance. For adjacent holes in the example, the distance is 2.1694" Next, draw a line from the center of the bolt circle to the midpoint if the previous line and write in half the measured distance (1.0847"). Two new equal right triangles are created by this line, each with half the total angle or 25.71º. The hypotenuse of the triangles is equal to the radius of the bolt circle and the side opposite the the 25.71º angle is half the measured distance. The side opposite the angle divided by the hypotenuse is equal to the sine of the angle. Rearranging, the radius of the bolt circle is equal to half the measured distance divided by the sine of the angle. If the measured distance between holes is a, the angle between holes is A, and the radius of the bolt circle is r, the bolt circle diameter, d, is d = 2*r = 2*a/2*sin(A/2) = a/sin(A/2) = a/sin(360/2*n) =a/sin(180/n)
This method can be generalized to measuring between any two holes. Again referring to the drawing, measuring the distance between holes that are three holes apart. The angle between the holes is 3 times the angle between adjacent holes or 154.29º and the half angle is 77. 14º. The measured distance is 4.8746" and half that is 2.4373". The same trigonometric relationships as in the previous example still hold. If m is the number of holes between the measured holes, the total angle is (m+1)*360/n and the half angle is (m+1)*180/n. The diameter now becomes
d = a/sin((m+1)*180/n).
This seems rather complicated but in reality, it is but a few key strokes on a scientific calculator. Using the latter example. enter 180 * 3/7 = 77.1428. Take the sine of that angle = .97493, invert it to get 1.02572, and multiply it by the measured center to center distance, 4.8746" to get 4.99996" for the bolt circle.
Now for coordinates for the holes. This is actually quite simple and straightforward.
Determine the angle between adjacent holes bu dividing 360 by the number of holes in the bolt circle. write that angle down. With the first hole at top (x=0, y =1), the next hole going clockwise is rotated by the angle between adjacent holes and the following hole rotated by that same amount or twice the angle from the top hole and so on for all holes where the angle is less than or equal to 180º. There is no need to go any further because the remaining holes are a mirror image of these. write the angles for the holes down. For each of those holes, take the sine of the angle. Multiply those values by the radius of the bolt circle. These are the x ordinates. Then take the cosine of those angles. Multiply those values by the radius of the bolt circle. These are the y ordinates.