Is the given shape a circle? What's the radius?

There are at least two ways a set of data can be checked to be points on a circle and determine the radius as well.

First, the radius of curvature can be measured using three consecutive points. This is a little advanced for all but second year calculus students. Another way is to assume that the points lie on a circle of a given radius . For this purpose it is most convenient to place the origin at the left most point of the circle. For example the circle of radius with center at the origin is written as but as a circle with left-most point passing through the origin it has the equation with graph shown below.

Continuing with an arbitrary radius and expanding we get which simplifies to . Solving for the radius we obtain

After obtaining our coordinates along the "suspected" circle we compute the value

for each

We confirm that the curve in question is a circle if all the values are about the same.

Here's and example:

• First import the image into Digitizer 1.0. You will obtain something like that shown below.
  
  

• Next define an origin (the red cross) and a point on the -axis. Specify its value.
  
  

• Now digitize points along the circle and show the data.
  
  

• The next step is to copy the data from the Digitizer data table into your spreadsheet and parse the data into columns. We show the pasted and then parsed data.
  
  

• The final step is the computation of the for each of the data points. Note that this method does not require multiple data points for a single computation.
  
  As is evident the data very nearly supports the hypothesis that the curve is indeed a circle. Note that the first point selected did not give accurate results. Why???

In summary, we required mathematics to use a proper model to test our hypothesis, the digitizer to make coordinates, and a spreadsheet to analyze the data.