Fire on the Mountain

Applications of planimeters and Green's Theorem

October 08, 2026 · John Peach

Almost ablaze still you don’t feel the heat
It takes all you got just to stay on the beat

• The Grateful Dead, Fire on the Mountain, Shakedown Street, 1978.

It’s more of a very small hill than a mountain, but the fire was real.

A little over one year ago, on July 6th 2025 Tropical Storm Chantal tore through central North Carolina, dumping more than 10 inches of rain here in Orange County, causing significant damage along Bolin Creek near our house as well as in Moore and Alamance counties (photo gallery). The ECONet ECONet is the Environment and Climate Observing Network, the weather-station network run by the State Climate Office of North Carolina at NC State. station near us recorded 7.65 inches of rain. Our friends Alan and Sue happened to be passing through that day, so they stopped for a visit. We thought they should have stayed overnight, but they pressed on and it turned out they were right. The heaviest rain was to the west of their route home and they quickly outran the worst of it. Another friend, Lisa, lost power so she spent the night with us.

Beginning in August, conditions changed radically. The area has been in a drought and by May 2026 we were in exceptional drought conditions. The US Drought Monitor has five levels, from D0 (abnormally dry) to D4 (exceptional). D4 is the worst.

Then, on May 20th a brush fire broke out in the Carolina North Forest near our house. The fire began Wednesday afternoon, with crews responding just before 4:30 p.m. The forest is maintained by the University of North Carolina, and we’re very fortunate to be able to hike the trails, but fire is always a worry especially during a drought. Fire departments from Chapel Hill and Carrboro arrived quickly, and coordination was turned over to the North Carolina Forestry Service. They were able to mostly contain the fire, and in an odd coincidence, it rained that night for the first time in months. Even so, in the days following the fire I saw open flames on fallen trees as I hiked through the forest.

How big was the fire? Clearly, it had been contained to a small patch of the forest, but I wanted to get a better estimate. I walked around the edges of the burnt area, recording my path with Strava Strava is a fitness app that records GPS tracks from a phone or watch. Under tree cover, GPS error can reach several meters. After getting a new phone I found that turning off power saving gives much more accurate GPS measurements . The paths acted as natural fire breaks so the area inside the loop burned, but there were no jumps across the paths except along the southern edge where fire crews bulldozed a new path.

To estimate the area inside the loop, we’ll look at Green’s Theorem and a curious device called a planimeter.

Planimeters

I grew up near the Chesapeake Bay in Maryland where many people owned sailboats and liked to race them on weekends. Every boat is unique, even when built by the same manufacturer because the amount of equipment carried changes the weight, sail sizes aren’t identical and many other factors affect performance. To compensate for these differences, formulas have been developed that penalize fast boats and give advantages to slower boats. After a race, times are adjusted according to the formulas and the boat with the fastest adjusted time is declared the winner.

My dad and I had a small business measuring boats for races. We’d go out on weekends and measure the overall length, waterline length, beam, sail areas and other factors that went into the racing formulas. Later, Maureen and I took over the measurement business and it earned us enough money to be able to go out to dinner and a movie. When we got home, we’d plug all the data into the formula to get the boat rating, but we also needed to estimate the wetted area, or the area below the waterline formed by the hull, keel and rudder.

Dad used his mechanical planimeter to find the wetted area. A planimeter has a pointer that follows the outline of the area and a couple of wheels that turn a dial. Start at one point, say $l_1$, follow the waterline back to $l_2$, then the dotted line around the rudder and keel and then the solid curve of the hull back to the starting point. After finishing the loop, you could read the dial to get the enclosed area. We’d then scale the result to the size of the boat and have an estimate of the wetted area. Planimeters may be either polar (shown in this photo) or linear, which we’ll discuss in detail below.

The Acme Planimeter

I needed a planimeter and found an online app called the Acme Planimeter. This sounds like something Wile E. Coyote would use to find the area the Roadrunner might be in. The Acme Planimeter shows a map of the world when it launches. Zoom in to the area you want to measure, and begin clicking points along the path until you’ve completed a loop.

I had to guess at the bulldozed path along the southern edge, but the other paths were clearly visible in the Acme map. The burned area was about $18 \pm 1$ acres (Acme estimated 17.59 acres) and since the entire forest is 833 acres this represents $2.1\%$ of the total that went up in smoke.

In high school during my boat measuring days I didn’t really think about how a planimeter worked, but as I was estimating the fire area I realized that it must be an application of Green’s Theorem.

Green’s Theorem

George Green was a largely self-taught English mathematician/physicist known for his essay on electricity and magnetism that he published in 1828. It wasn’t until he was 40 years old that he entered the University of Cambridge where he wrote papers on acoustics, hydrodynamics and optics.

Green’s Theorem relates the vector field A vector field puts an arrow, with a direction and a size, at every point, like the wind arrows on a weather map. inside a closed area to the field at the edges. Imagine you’re in a swimming pool with a bunch of whirlpools swirling around.

The whirlpools are the white curved arrows and the red arrows are the vector field. The vector field shows the direction and speed of the water flowing at each point. I’ve only shown the vector fields around the middle two whirlpools, but the same flow applies to all of them. Every whirlpool turns counterclockwise, so if you swim along the edge of the pool, the water pushes you counterclockwise too.

Green’s Theorem Cauchy published the planar form in 1846 and Riemann first proved it, though it carries Green’s name. says that the flow at the edge tells you something about the flow inside the pool, and vice versa. Look at the center of the pool. The whirlpool on the left sends water upward and the whirlpool on the right sends it downward, so the two flows oppose each other and the water at that spot barely moves. That describes the velocity at one point. The spin is a separate matter: each whirlpool keeps spinning, and the spins from different whirlpools add up.

To see why the edge measures that total spin, cut the pool into a grid of tiny square cells. Walk around one cell counterclockwise and add up how much the water helps you along the way. This sum is the cell’s circulation, and for a tiny cell it is proportional to how fast the water spins there. Two neighboring cells share an edge, and you walk it in opposite directions for the two cells. The push along that edge counts once as a gain and once as a loss, so it cancels. After all the cancellations, the only edges left are those on the outer boundary of the pool. The circulation around the pool’s edge therefore equals the sum of the circulations of all the cells, which is the total spin inside.

The edge tells you the total spin but nothing about where it is. Three whirlpools crowded into one corner and the same three spread across the pool give the same reading at the edge.

If you add up all the microscopic swirling inside the region (a 2D area sum), the internal motion along edges shared by neighboring cells cancels, leaving only the net fluid flow running along the outer perimeter (a 1D loop path). Green’s Theorem states that adding up the internal local spin across an entire area equals moving around the boundary loop and measuring the total circulation Circulation is the total push a flow gives you along a loop. Curl is the local version: how fast a tiny paddle wheel at that point would spin. .

The mathematical statement of his theorem is

where $\oint_C$ means the integral around the boundary and $\iint_D$ is the integral over the domain (interior). In terms of the vector field,

where

• $L(x, y)$ is the horizontal ($x$-direction) component of the vector field at any point $(x, y)$,
• $M(x, y)$ is the vertical ($y$-direction) component of the vector field at any point $(x, y)$,

and $\bold{i}$ is the vector along the $x-$axis and $\bold{j}$ points along the $y-$axis.

The right side of the equation describes the interior. Instead of moving around an entire swimming pool, suppose you travel around an infinitesimally small square with sides $\Delta x, \Delta y$.

Move in the $x-$direction by a distance $\Delta x$, and the work done by the water flowing is the force $L(x,y)$ times the distance $\Delta x$. Along the bottom edge, this is

while the same holds at the top in the opposite direction, so

which has a minus sign because you’re moving from right to left. The sum of the two horizontal forces over the distance $\Delta x$ becomes,

The vertical component is similar, except the signs change

The total work done around the square is then,

How Green’s Theorem measures area

Looking at Green’s Theorem again,

the area inside the closed curve $C$ is

so we want a “flow” such that

The easy choice for this is to set $M = x$ and $L = 0$, so the area becomes

Suppose you want to measure the area of the rectangle with points $(0,0),(3,0),(3,2),(0,2)$, which is $3 \times 2 = 6.$ Starting at the origin, travel counterclockwise along the $x-$axis from $0$ to $3$, then up to $(3,2)$ and so on. The integral $\oint x \; dy$ only counts motion in the $y$-direction, so the horizontal edges add nothing. The left edge has $x = 0$, so it adds nothing either. The right edge has $x = 3$ and length $2$, so it adds $3 \times 2 = 6$, the area of the rectangle.

From the other side of the equation,

so the area is the double integral of $1$ over $3 \times 2 = 6$.

Another, more symmetric version sets $M = \frac{1}{2}x$ and $L = -\frac{1}{2}y$. Here, the curl is $\frac{1}{2} + \frac{1}{2} = 1,$ so the area is

How the planimeter measures area

The linear planimeter in the photo has two carriage wheels on a common axle. The carriage rolls forward and back on them, and the frame carrying the tracer arm slides along the axle. The stylus can reach any point on the paper by combining the two motions. Choose coordinates so that $x$ is the direction the carriage rolls and $y$ is the direction along the axle.

The two motions are recorded separately. When the carriage rolls a distance $dx$, the wheels turn the axle, and the axle turns a flat integrating disc through a small angle

where $\kappa$ is a constant set by the wheel size and the gearing between the axle and the disc. When the stylus moves in $y$, the frame slides along the axle and carries a small measuring roller across the face of the disc. The roller rests on the disc at a distance $\rho$ from the disc’s center, and $\rho$ is proportional to $y$. I’ll measure $y$ from the position where the roller sits at the exact center, so $\rho = y$.

A roller resting on a spinning disc is dragged around by it. At distance $\rho$ from the center, a disc turning through $d\alpha$ moves its surface a distance $\rho \; d\alpha$ under the roller, so the roller rolls

Rolling distance is radius times angle, which is the same fact that lets a bicycle wheel’s odometer measure distance. The roller feeds the counter drum and the dial scale, which record the total rolling distance $w$.

Each kind of stylus motion shows up differently on the dial:

• Moving in $x$ at large $y$ spins the disc under a roller far from its center, so the roller rolls a long way.
• Moving in $x$ at small $y$ rolls it only a little.
• Moving in $x$ at $y = 0$ puts the roller at the center of the disc, which turns without moving the roller’s contact point at all.
• Moving in $y$ alone slides the roller across a stationary disc, so it slides without rolling.

Now trace a closed curve $C$ and let the roller accumulate $dw$ along the way:

The measuring roller with radius $a$ turning through $N$ revolutions rolls $2\pi a N$, so the dial reading gives

Where you put the zero of $y$ doesn’t matter. Shifting it by a constant $y_0$ adds $y_0 \oint dx$ to the integral, and $\oint dx = 0$ for a closed curve because $x$ ends where it started.

You can check this on the rectangle from the previous section, with corners $(0,0)$, $(3,0)$, $(3,2)$, $(0,2)$, traced clockwise. On the two vertical sides $dx = 0$, so the disc doesn’t turn and the roller only slides. On the bottom edge $y = 0$, so the roller sits at the center of the disc and doesn’t roll. On the top edge $y = 2$ and $x$ increases by $3$, so the integral is $2 \times 3 = 6$, the area of the rectangle.

Relation to Green’s Theorem

The planimeter computes $\oint y \; dx$, which is Green’s Theorem with a particular choice of field. Setting $L = -y$ and $M = 0$ gives

so the curl is $1$ everywhere and

for a counterclockwise loop. A clockwise trace flips the sign, which is why it gives $+\oint y \; dx$. This is the same trick as $M = x,\ L = 0$ from the previous section, using a different field.

In terms of the swimming pool, the field $-y \; \mathbf{i}$ is a shear flow. The water moves left above the line $y=0$ and right below it, with speed proportional to the distance from the line. Every small patch of this flow spins at the same rate, so the total spin inside the loop is proportional to its area. Green’s Theorem says that equals the circulation around the edge, and the disc and roller are a mechanical device for adding up that circulation.

Proof of Green’s Theorem

The proof is two applications of the fundamental theorem of calculus Adding up a rate of change gives the net change: $\int_a^b f'(x) \; dx = f(b) - f(a)$. , once in $y$ and once in $x$. I’ll prove it for a simple kind of region, then show how any other region is built from simple pieces. I assume $L$ and $M$ have continuous partial derivatives, so every integral below exists.

Green’s Theorem says

Each side is a sum of an $L$ part and an $M$ part, so it’s enough to prove two smaller statements:

Adding them gives the theorem.

The $L$ half

Suppose the region $D$ lies between a bottom curve $y = g_1(x)$ and a top curve $y = g_2(x),$ for $x$ running from $a$ to $b$. Walk around the boundary counterclockwise: along the bottom from left to right, up the right side, back along the top from right to left, and down the left side.

On the two vertical sides $x$ doesn’t change, so $dx = 0$ and they contribute nothing to $\oint L \; dx$. That leaves the bottom and the top:

The top has a minus sign because we walk it from right to left.

Now let’s look at the double integral. Integrate in $y$ first, and let the fundamental theorem of calculus do the inner integral:

This is the negative of the boundary integral above, which proves

The planimeter is a special case. With $L = -y$, the bottom curve contributes $-\int_a^b g_1 \; dx$ and the top contributes $+\int_a^b g_2 \; dx$, so the loop integral is $\int_a^b (g_2 - g_1) \; dx$. That is the area between the curves, which is what the dial reports.

The $M$ half

The same steps work with $x$ and $y$ swapped. Describe the region as lying between a left curve $x = h_1(y)$ and a right curve $x = h_2(y)$, for $y$ running from $c$ to $d$. Walking counterclockwise, we go up the right curve and down the left curve. The horizontal pieces have $dy = 0$, so

Integrating in $x$ first, the fundamental theorem of calculus gives

The two expressions match, so $\oint_C M \; dy = \iint_D \frac{\partial M}{\partial x} \; dA$. There is no minus sign this time because the right side is walked in the direction of increasing $y$. That sign difference between the two halves is why the theorem has $\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y}$ rather than a sum.

Regions that aren’t simple

The argument above needs a region that can be described both ways: between two curves of $x$, and between two curves of $y$. Rectangles, discs, and triangles qualify. A crescent or an L-shaped region may not.

For those, cut the region into simple pieces and apply the theorem to each piece. Every cut is the edge of two neighboring pieces, and each piece walks it in the opposite direction, so the two line integrals cancel. The double integrals over the pieces add up to the double integral over $D$. Only the original boundary is left on the line-integral side, and the theorem holds for the whole region.

Look at the swimming pool again. The swirling inside the region cancels between neighbors, and only the flow along the outer edge survives. A region with a hole works the same way. Cut from the hole to the outer edge, and the hole’s boundary comes out traversed clockwise.

Conclusion

The loop I walked around the burned area in the Carolina forest was about 18 acres. The Acme planimeter shows 17.59 acres, but I had to guess the path of the newly bulldozed line. I used the Acme planimeter to measure the total area of the forest which gave 833 acres. My memory of Dad’s planimeter led me to thinking about how it measured area which got me thinking about Green’s Theorem. It says that sum equals the area inside, so the boundary holds the whole answer. In the disc-and-roller planimeter the piece is $y \;d x$, and in the symmetric version it is $\tfrac{1}{2}(x \; dy - y \; dx)$. Each step along the edge adds its piece, and the dial or the app reports the total.

If you recorded a track of your own loop, you can skip the clicking. A track is already a list of GPS points, and you can sum the symmetric formula along each straight segment between them. Convert latitude and longitude to meters first, because a degree of longitude gets shorter as you move north. Compare your answer with the one from clicking. The gap between the two gives you a feel for how much to trust either one. You could also nudge every point a few meters in a random direction, rerun the sum a few hundred times, and see how much the area moves.

If you’d like to experiment with building a planimeter Robert Foote and Ed Sandifer show how to do it with Tinker Toys in Area Without Integration: Make Your Own Planimeter which also gives an excellent explanation of how they work. If you have a 3D printer, you could build one following Alden Bradford’s instructions in A 3D-printed Planimeter For Use In Classrooms.

References

• Edwards, B. H. (2007). “The planimeter.” Revista de Matemática: Teoría y Aplicaciones 14(2), 177–182. It derives the polar planimeter from Green’s Theorem.
• R. Foote and E. Sandifer Area Without Integration: Make Your Own Planimeter
• A. Bradford, A 3D-printed Planimeter For Use In Classrooms, July 24, 2022.
• Mathematical Instruments, Planimeters for measuring area.
• B. McConnell, Brush Fire Contained, But Still Burning in Carolina North Forest; Officials Warn of Smoky Conditions, Chapelboro, May 21, 2026.
• Smoky Conditions Continue After Brush Fire, Town of Chapel Hill, May 20, 2026.
• C. Davis, Rapid Reaction: Tropical Storm Chantal Soaks Central North Carolina, North Carolina State Climate Office, July 7, 2025.

Image credits

• Brush Fire, Chapelboro News, May 21, 2026
• U.S. Drought Monitor for North Carolina
• Strava map of the fire perimeter
• Mechanical Planimeter, By Schorle - Own work, CC BY-SA 3.0
• Acme Planimeter