What is Area = πr² for clover?

In 1990, the second year students in Agriculture at The University of Melbourne measured several things about a sample of 140 3-leafed clovers, including:
Midrib , the length of the spine running through the middle leaf, in cm, and
Area , the actual area of the entire clover (all three leaves), in cm².
clover leaf How can the midrib length be used to predict the area of the clover?

First, here's a graph of area versus midrib:

clover leaf

Clovers with longer midribs have greater area. However, the relationship is curved, and the accuracy of prediction is worse for larger clovers. This pattern of increasing variation for larger values is common and an often be corrected by taking logarithms of the data, and analysing the results.

Here's the result.

Leaf2

The relation is quite linear.  The inaccuracy of a straight line prediction no longer increases as it did in the original plot.

We can consider a straight line model for these data.  More formally this is called a linear regression model.  It takes the form: log(area) = a + b log(m) + error, where m is the midrib.  The error describes the unpredictable or random variation from the line.

Here's the graph with the line.

Leaf3

Using a statistics package, we can estimate a and b.  This gives the equation log(area) = .73 + 1.80 x log(m).

Transforming this relation back to the original variables (by reversing the logarithms) gives the prediction curve shown here:

leaf4

The equation of this curve is: area = 2.08 x m 1.8 . We can rewrite this equation in terms of half of the midrib, called r. The equation becomes: area = 7.24 x r 1.8 . Remember the area is for three leaves, so we could write: area = 3 × 2.41 x r 1.8 .

Now for circles, the relation between area and radius is area = π x r 2 . The analogue for clovers is Area = 3 × 2.41 x r 1.8 .

The power of the radius, 1.8, is not too different from 2, but the area increases a bit more slowly with radius for leaves than for circles.

Conclusion: The relation Area = 3 × 2.41 r 1.8 is the analogue for these clovers to Area = πr 2 ; for circles.