Measuring Area by Walking the Edge

A planimeter is a mechanical tool that measures the area of any shape on a map just by tracing its outline. It works because of Green's theorem: a line integral around the edge equals a double integral over the inside, and the right field makes that double integral the area.

Click or drag anywhere to start a new trajectory there.

How it works

Green's theorem

In one sentence

If $C$ is a positively oriented, piecewise smooth, simple closed curve bounding $D$ , and $P$ , $Q$ have continuous partials on $D$ , then $\oint_C P\,dx + Q\,dy = \iint_D (Q_x - P_y)\,dA$ ; with $P = -y/2$ , $Q = x/2$ the right side is the area of $D$ .

General formula

\text{Area}({\htmlData{sym=1}{D}}) = {\htmlData{sym=0}{\oint_C}} \left(-\frac{\htmlData{sym=4}{{y}}}{2}\,d{\htmlData{sym=3}{x}} + \frac{\htmlData{sym=3}{{x}}}{2}\,d{\htmlData{sym=4}{y}}\right) = \iint_{\htmlData{sym=1}{D}} 1\,{\htmlData{sym=2}{dA}}

Curl measures how much a field spins at each point. This field spins the same amount (1) everywhere, so the total spin inside a loop, which the loop's circulation measures, is just the area.

What the symbols mean

$\oint_C$
Around the edge
Add up along the shape's boundary C, counterclockwise.
$D$
The region
The shape whose area you want.
$dA$
Area element
A tiny piece of the inside.
$x$
Horizontal position
Where the tracer is on the map.
$y$
Vertical position
Where the tracer is on the map.

Green's theorem: $\oint_C P\,dx + Q\,dy = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)dA$ .
Choose $P = -y/2$ and $Q = x/2$ . Then $Q_x - P_y = \tfrac12 + \tfrac12 = 1$ , so the double integral is just the area.
So tracing the edge and accumulating that line integral measures the area inside, for any shape.
A planimeter's wheel performs exactly this integration as its arm follows the outline.

Learn the math

Green's TheoremCalc III · Lesson 44, Line Integrals