# Measuring Areas with a Shopping Cart

**Figure 1.**Trajectories of the two wheels as the front one repeatedly circumscribes a closed curve.

\[A= \theta L ^2 + O( \epsilon ^3 ), \tag1 \]

where \(L\) is the length of the "bike."

**Figure 2.**The hatchet planimeter; (1) gives the area.

\[z\mapsto e^{i \alpha } \frac{z-a}{1-\overline a z},\]

**Figure 3.**

*RF*is a directed segment with

*R*’s velocity constrained to the line

*RF*.

A geometrical explanation of \((1)\) rests on two observations of independent interest.

**Figure 4.**The doubly-swept area contributes zero, leaving

*A*as the net swept area.

_{Q}-A_{P}### Observation 1

The signed area** ^{1}** swept by a moving segment (as described in Figure 3) remains unchanged if the longitudinal velocity is altered (and in particular made to vanish).

This statement is intuitively plausible since the longitudinal motion has no effect on the rate at which \(RF\) sweeps the area.

### Observation 2

A segment \(PQ\) executing a cyclic motion in the plane sweeps the signed area \(A_Q-A_P\) (see Figure 4).

**Figure 5.**Proof of Prytz’s formula (1).

### Proof of Prytz’s Formula (1)

Figure 5 shows the motion of \(RF\) over one zig-zag, with \(F\) returning to its starting position; the rotation around the start/stop point through \(\theta\) completes the cyclic motion, bringing \(RF\) to its initial position (the rotation violates the no-slip condition).

During the “sliding” stage, \(RF\) sweeps area \(\frac{1}{2} \theta L ^2\), according to Observation 1. During the “rotating” stage, \(RF\) sweeps the sector of area \(\frac{1}{2} \theta L ^2\). The total swept area is thus \(\theta L ^2\).

But this area equals \(A -A_R\) by Observation 2, so that

\[A -A_R= \theta L ^2.\]

**Figure 6.**

*πc*+ring=

^{2}=πa^{2}*πa*.

^{2}+πb^{2}It is worth noting that if one extends the “bike” to \({\mathbb R} ^3\), Prytz’s formula \((1)\) admits an eye-opening explanation entirely different from the one I just described. This explanation (given in [1]) is similar in spirit to the explanation of the finiteness of the radius of convergence of \(1/(1+ x ^2 )\) by extending to the complex domain.

As a concluding remark, choosing a circular annulus in Figure 3 yields another proof of the Pythagorean theorem (modulo the proof of Observation 1), as Figure 6 illustrates.

The figures in this article were provided by the author.

** ^{1}** The area counts with the positive sign if the motion is to the left of \(RF\), as it is in Figure 3.

**References**

[1] Bor, G., Levi, M., Perline, R., & Tabachnikov, S. (2017). Tire tracks and integrable curve evolution. Preprint,

*arXiv:1705.06314*.

[2] Foote, R. (1998). Geometry of the Prytz planimeter.

*Rep. Math. Phys., 42*, 249-271.