The Birkenhead projectMeasuring Birkenhead Park from its outline
Download a polygon, turn degrees into metres, and compare with the acres in an old book.
OpenStreetMap stores the park as a single closed outline of 181 points, each a latitude and longitude. The shoelace formula gives the area of any such polygon: multiply each point's x by the next point's y, subtract the reverse, add everything up and halve it. Run it straight on the coordinates and the answer is 0.0000779, in square degrees, a unit that means nothing on the ground. The learner has to project the points onto a flat grid in metres first, and that is where the real lessons are.
The area of Birkenhead Park by method, our Python run on OpenStreetMap way 257607585, 28 September 2026| Method | Result | Acres | Verdict |
|---|
| Shoelace on raw degrees | 0.0000779 square degrees | None | Wrong units |
| Degrees to metres, no latitude correction | About 963,000 square metres | 238.0 | Far too big |
| Local projection with cos(latitude) | 574,279 square metres (sign negative) | 141.9 | Close |
| Geodesic area on the WGS84 ellipsoid | 576,688 square metres | 142.5 | Reference |
| Olmsted, 1850 visit: park and garden | One hundred and twenty acres | 120 | Quoted |
| Olmsted, later article | A piece of ground of 185 acres | 185 | Quoted |
A degree of latitude is about 111 kilometres everywhere, but a degree of longitude shrinks towards the poles. At Birkenhead, about 53.4 degrees north, it is only about 60 percent as long, so east-west distances must be multiplied by the cosine of the latitude. Forget that step and the park comes out at 238 acres, 68 percent too big. With the correction the local projection gives 574,279 square metres, which is 141.9 acres at 4,046.8564224 square metres to the acre. The sign of the result is negative because the outline runs clockwise, a detail that trips up anyone who does not take the absolute value.
As a check, the learner uses the geographiclib package, which measures the polygon on the true shape of the Earth rather than a sphere: 576,688 square metres, or 142.5 acres, less than half a percent from the simple method. So the modern outline encloses about 142 acres. That is more than the 120 acres of park and garden the gardener described in 1850, and less than both the 185 of Olmsted's later article and the 180 implied by his own "remaining sixty acres" set aside for building plots. The numbers do not settle who was right, because each describes a different boundary; the program's job is to measure honestly and say which boundary it measured.
Ages 8 to 11
Draw the park on squared paper, count the squares and compare with a friend's count.
Ages 11 to 15
Code the shoelace formula in Python and test it on shapes with known areas.
Ages 15 and up
Project the real outline, handle orientation, and compare with a geodesic library.
Olmsted's words, open map data, our sums
The quotations come from the Project Gutenberg edition of Frederick Law Olmsted, Landscape Architect (1922). The park outline is OpenStreetMap data, © OpenStreetMap contributors, available under the Open Database Licence. Every area in the table is our own calculation.
