Geometric Measurement With Differentials

One Radius, Two Different Measurement Effects

A small uncertainty in a circle’s radius affects both its circumference and its area. But the resulting absolute uncertainties do not have the same scale: the circumference estimate depends on the radius uncertainty alone, while the area estimate also depends on the circle’s radius.

Differentials make this comparison direct. They describe the first-order change in an output produced by a small change in an input. For circumference, that first-order estimate is exact. For area, it is a local estimate: the exact change also contains a second-order term.

Circle Measurements and Their Differentials

Let \(r>0\) be a circle’s radius. Its circumference \(C\) and area \(A\), as functions of the radius, are

Suppose the measured radius is \(r_0\), and its actual value differs by \(\Delta r\). Thus the actual radius is \(r_0+\Delta r\). The differential uses a small input change \(dr\); when estimating the effect of the radius uncertainty, we set \(dr=\Delta r\).

Differentiate each circle formula with respect to \(r\). The resulting differentials are

For an uncertainty estimate based at the measured radius \(r_0\), use that base point in the area differential:

The units provide a useful check. If \(r\) and \(dr\) are measured in centimetres, then \(dC\) is in centimetres, while \(dA\) is in square centimetres. The circumference estimate depends on the size of \(dr\), but not on \(r_0\). The area estimate depends on both.

Why the Two Estimates Differ

The distinction follows directly by comparing the differentials with the exact changes. If the radius changes from \(r_0\) to \(r_0+\Delta r\), the exact circumference change is

With \(dr=\Delta r\), this is exactly \(dC\), not merely an approximation. This happens because \(C(r)\) is a linear function of \(r\).

For area, expand the square to find the exact change:

The differential \(dA=2\pi r_0\Delta r\) is the first term. The remaining term, \(\pi(\Delta r)^2\), is second order in the radius change. It is positive for every nonzero \(\Delta r\), whether the radius increases or decreases. This explains why the differential gives a useful estimate for small changes but is not generally the exact area change.

Worked Examples

Absolute and Relative Uncertainty

Absolute uncertainty is expressed in the output’s units. The differential estimates show that for a radius uncertainty of magnitude \(\delta\), the circumference uncertainty is \(2\pi\delta\), while the area uncertainty is approximately \(2\pi r_0\delta\). Thus a fixed absolute uncertainty in radius gives a fixed absolute circumference uncertainty, but a larger estimated absolute area uncertainty for a larger circle.

Relative uncertainty compares an output change with the output itself. Dividing the differentials by the corresponding measurements gives, for small changes,

So the estimated relative circumference change matches the relative radius change, while the estimated relative area change is twice the relative radius change. These are relative first-order estimates; the area’s exact change still includes the second-order term. Absolute and relative uncertainty answer different questions, so they should not be interchanged.

A Common Mistake: Squaring the Radius Uncertainty

A frequent error is to treat the area formula as though its differential were obtained by simply replacing \(r\) with \(dr\). For a circle with \(r_0=5\text{ cm}\) and \(dr=0.1\text{ cm}\), that incorrect calculation gives

This is not the differential. Differentiate \(A(r)=\pi r^2\) first: \(A'(r)=2\pi r\). Then evaluate at the measured radius and multiply by the input change:

The exact increase for this radius change is \(\pi(5.1^2-5^2)\text{ cm}^2=1.01\pi\text{ cm}^2\), which is close to the differential estimate \(\pi\text{ cm}^2\). The incorrect value \(0.01\pi\text{ cm}^2\) misses the first-order effect. The correction is to differentiate the measurement formula, then evaluate its derivative at the base radius.

Using the Estimates Responsibly

Differentials are especially useful when the radius uncertainty is small compared with the radius itself. In that setting, \((\Delta r)^2\) is small relative to the first-order term \(2r_0\Delta r\), so the area differential is a good estimate. When an exact bound is needed, use the endpoint radii \(r_0-\delta\) and \(r_0+\delta\) in the area formula, provided \(\delta<r_0\). The resulting area uncertainty is generally asymmetric around the measured area.

For circumference, no such approximation is needed: its exact change is already given by the differential. For area, the differential captures the leading effect and makes the dependence on the measured radius visible. These different scales matter whenever a geometric measurement is reported from an uncertain radius.