Why aren't four sides enough?
On a rectangular lot, the area is length × width. But if the lot's corners aren't right angles, the four side lengths don't determine the area: the corner angles can change while the sides stay the same, and the area changes with them. Both shapes below have four 20 m sides; one is 400 m², the other about 346 m².
That's why, on a four-cornered lot, you need to know a diagonal (the distance between two opposite corners) in addition to the four sides, or else the positions of the corners. The more corners there are, the more lengths you need to measure.
Method 1: Split into triangles with a diagonal (Heron's formula)
A triangle with three known sides has only one possible area. If you split the lot into two triangles with a diagonal, you calculate each triangle's area from its sides and then add them up. Our example lot is ABCD; its sides and the diagonal AC were measured with a tape.
Heron's formula gives a triangle's area from its three sides: first find the semi-perimeter, s = (a + b + c) ÷ 2. Area = √(s × (s − a) × (s − b) × (s − c)).
| Triangle and sides | Semi-perimeter (s) | Area |
|---|---|---|
| ABC24.00 · 18.44 · 26.91 m | 34.675 m | ≈ 216.0 m² |
| ACD26.91 · 18.25 · 15.13 m | 30.145 m | ≈ 132.0 m² |
| Site | ≈ 348.0 m² | |
Let's write out triangle ABC: √(34.675 × 10.675 × 16.235 × 7.765) ≈ 216.0 m². On a lot with five or more corners, keep going the same way: draw diagonals from one corner to all the opposite corners; a lot with n corners splits into n − 2 triangles. If the lot has an inward-pointing corner, choose the diagonals so that they run inside the lot.
Method 2: Corner coordinates (shoelace formula)
If you know where the corners are, from a map, a survey table or by measuring from one corner taken as the origin, there's a shorter way. Write the corners in order around the lot's boundary (clockwise or counterclockwise, but without skipping any). On each row, multiply this corner's x by the next corner's y, then subtract the next corner's x times this corner's y. That's the “cross product”. Add up the rows (here 696); half of the absolute value is the area. This method is called the shoelace formula, also known as Gauss's area formula.
| Corners | x (m) | y (m) | Cross product |
|---|---|---|---|
| A | 0 | 0 | 0 × 0 − 24 × 0 = 0 |
| B | 24 | 0 | 24 × 18 − 20 × 0 = 432 |
| C | 20 | 18 | 20 × 15 − 2 × 18 = 264 |
| D | 2 | 15 | 2 × 0 − 0 × 15 = 0 |
| Total | 696 ÷ 2 = 348 m² | ||
Both methods give the same result; the small difference comes only from rounding the measurements to the centimeter. The coordinate method works with the same table for any number of corners; the only condition is that the boundary lines don't cross each other.
A true trapezoid: two parallel sides
In geometry, a trapezoid is a four-sided shape with two parallel sides. If the front facing the street and the back boundary are parallel, there's a shortcut: add the two parallel sides, divide by two and multiply by the perpendicular distance between them.
The height here is not the length of the side; it's the perpendicular distance between the two parallel sides. If the side is slanted, the height is shorter than the side.
What to watch out for when measuring
- Mark the corner points: a boundary stone, a wall corner or a stake. Measure every length between the same two points.
- Don't forget the diagonal: at least one diagonal with four corners; with more corners, from one corner to all the opposite corners.
- Keep the tape straight and taut: a sagging tape makes lengths look longer; measure a long side by splitting it with intermediate points.
- Sloping land: the area on maps and in official records is usually calculated on the horizontal plane. If you lay the tape on the ground along a slope, the sides come out slightly too long; keep the tape horizontal.
- Check your work: measure the second diagonal too and compare the two methods. If the difference is large, one measurement is wrong.
Lot area in Nodaform
In Nodaform's Site step, you draw the lot corner by corner; the area and perimeter are calculated automatically from the corners.
- Click the corners in order: corners snap to the grid, Shift locks the angle, and you can type the side length. Clicking the first corner closes the lot.
- Adjust: drag corners, add a new corner with the point in the middle of a side; compare the side lengths in the panel with your own measurements.
- In the panel: number of corners, perimeter, each side's length and the lot area; once you draw the house, the coverage ratio. The distances from the house to the plot boundary appear on the plan.
Turn your phone sideways and you can place the corners by tapping, too. The Site step is in every plan, including Free.
This article is for general information. For buying, selling, title deed and zoning matters, the area in the official record applies; for the exact dimensions and boundaries of a lot, consult a licensed surveyor or the relevant authority.