These are attachments for a 6" Mitutoyo digital caliper for measuring radii. Any size radius is supported, and the two different formulas are printed on the larger attachment, as well on the reference target block.
Both the large and the small attachment are designed to allow for measuring any arbitrary-sized radius, using one of the following basic formulas. Accuracy gradually decreases for radii where a 90° arc segment has a chord larger than the max chord of the tool.
Variables:
r = radius
m = measured distance on the caliper
c = max chord of the tool at the tips of the attachment
l = ½ of the maximum chord (c * 0.5)
s = sagitta, aka height of the arc (l - m)
For a 90° arc segment that has a chord of less than 100mm (small attachment) or 200mm (large attachment), the arc fits inside the tool, so the simple formula can be used:
r = m * 2.4142 (rounded value)
Example: measure the 25mm radius, and measure a value ~10.36mm, then multiply by 2.4142 to get 25mm.
25 = 10.36 * 2.4142
For large radii that will not fit inside the tool, the radius can still be determined using one of the following formulas:
r = (s² + l²) / (2*s)
or
r = c² / (8*s) + s / 2
We'll use the first formula in these examples.
For the small attachment, the chord is 100, so half the chord (l) is 50. For the large attachment, the chord at the tips is 200, so half is 100.
50² = 2500 and 100² = 10000
So for the small attachment:
s = 50mm - m
r = (s² + 2500mm) / (2*s)
For the large attachment:
s = 100mm - m
r = (s² + 10000mm) / (2*s)
In Practice:
If you have a measurement of 40mm, then with the small attachment, the s = 10, so:
r = (10² + 50² ) / (2 * 10)
r = (100 + 2500) / (20)
r = 2600 / 20
r = 130mm
If you have a measurement of 80mm with the large attachment, the s = 20, so:
r = (20² + 100² ) / (2 * 20)
r = (400 + 10000) / (40)
r = 10400 / 40
r = 260mm
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