Finding a Radius

FINDING A RADIUS

by

James R. Howe

(6/12/2001)

This article was developed to solve the problem of how to measure a radius that is less than a full diameter. When it is necessary to inspect the diameter of a bore we can use a dial caliper, an inside micrometer, or a telescope gage and outside micrometer. The method is simple enough as the measuring device can easily span the diameter. But what do you use when the radius to be measured is less than a full diameter? We cannot use a CMM because we don't have one. The photo readily shows the dilemma.

The radii are marked with red arrow. It is easily seen that the common measuring tools mentioned above will not work, there is not a complete diameter. Since it is imperative to measure these radii for wear. The following technique is recommended.

r = (c2 + 4h2 ) / 8h *

Arc

c h

r

Works like a charm! However, be careful with your "c" and "h" measurements.

Procedure: After tracing the radius on a suitable surface (a 3x5 index card works well for radii under 6.00") carefully draw line "c" using a 6" metal rule. Choosing a dimension that is easily divisible by two allows a quick midpoint setting of "c". Using the same 6" metal rule carefully draw a perpendicular line "h" from the midpoint of "c" to the "arc". Using a dial caliper carefully measure "h" from the midpoint of "c" to the very outside of the "arc". Now that the dimensions are known use a calculator to determine "r". Multiply "r" by two for diameter.

To check your calculation set the dial caliper to the calculated "r" value. Using the 6" metal rule extend line "h". Place the dial caliper on "h" with the very tip of the caliper jaw on the outside of the "arc". Rotate the caliper and observe how closely it follows the "arc" from the one intersection with "c" to the other intersection of "c". It should follow exactly. If it does not then your calculations and/or measurements are probably in error.

The measured values can change dramatically so be careful with your technique. (see below discussion on ERRORS)

J.R. Howe

QA-Akron

*The Equation is from: The MACHINERY'S HANDBOOK 23

ERRORS IN THE PROCEDURE FOR FINDING A RADIUS

r=(c2+4h2)/8h

Beginning with a real world radius of 1.7745 or diameter = 3.549 (optimum of 3.547/3.551) for a SK330-6 slide-loc. We can set c at 2.5 and hold it while we vary h. we can then determine how sensitive the equation is when h varies. To do this we must know what h is under optimum conditions.

If r=1.7745, and c=2.5, then we can solve the equation for h:

r=(c2+4h2)/8h

1.7745=(6.25+4h2 )/8h

14.196=(6.25+4h2)/h

14.196h=6.25+4h2

0=6.25+4h2-14.196h

0=4h2-14.196h+6.25

or 0=h2-3.549h+1.5625

This is a quadratic equation solvable by using

x=-b +/- (b2-4ac)1/2 / 2a

x={3.549 +/- [(3.5492)-(4x1x1.5625)]1/2 } / 2x1

x={3.549 +/- (6.3454)1/2 } / 2

x=(3.549 +/- 2.519) /2

x= 1.03/2 and/or x= 6.06/2

x= .515 and/or x= 3.034

Since the 3.034 does not fit in our radius world we take the .515 as optimum h. We can now plot r for each value assigned to h +/- .001 increments from the optimum .515.