What it does
Real plots of land are rarely rectangles. Surveyors and farmers find their area the way you can at school: cut the plot into triangles, measure the three sides of each triangle, and use Heron’s formula, area = √[s(s − a)(s − b)(s − c)] with s half the perimeter. This page does that live while you drag the corners:
- the plot is drawn on a grid of 10 m squares, with every side’s length marked,
- it is cut into triangles T1, T2, … that stay inside the plot, even when the plot has a dent,
- each triangle’s sides, s and Heron area are written out, then added up,
- the total is given in m², hectares and acres, with the perimeter (the fence you would need),
- at the Class 11 level the corners are given as coordinates and the shoelace formula gives the same area in one line.
It follows the Class 9 maths chapter on perimeter and area (Heron’s formula for triangles and fields), and links to Class 11 coordinate geometry.
For a single triangle, step by step, use heron's formula explorer; the area of a triangle from its corner coordinates is in the section formula tool, and four-sided shapes are explored in the quadrilateral explorer.
How to use it
- Drag any corner. Corners snap to whole metres; the area and the triangles update as you go.
- Add a corner puts a new one in the middle of the longest side; tap a corner to select it, then Remove a corner.
- Try the quick plots: a five-sided field, an L-shaped plot (with a dent) and a triangle.
- Read Triangle by triangle for the full working. Switch to Class 11 for the coordinates and the shoelace check.
- If two sides cross, the page says so: a crossed shape does not have a proper inside.
A worked example
The five-sided field has corners A(10, 10), B(110, 10), C(150, 90), D(70, 140) and E(20, 100), in metres. It splits into 5 − 2 = 3 triangles:
- EAB: area 4 500 m² (sides 90.6, 100 and 127.3 m),
- EBC: area 5 400 m²,
- CDE: area 2 850 m².
Total: 4 500 + 5 400 + 2 850 = 12 750 m², which is 1.275 hectares or about 3.15 acres. The perimeter is about 438.4 m. The shoelace formula gives the same 12 750 m².
Tips
- Any polygon with n straight sides can be cut into n − 2 triangles by lines joining its corners.
- Heron’s formula needs only the three side lengths, which is why it suits fields: you can measure sides with a tape, but heights are hard to measure on the ground.
- 1 hectare is a 100 m by 100 m square. An acre is a little over 40 % of a hectare.
Check yourself
Four fresh questions each time: a Heron triangle, a rectangle with a triangle on top, m² to hectares, and how many triangles a polygon splits into. Nothing is saved.
FAQ
Why do the triangle areas sometimes end in long decimals? Side lengths along a slant are square roots, so the working is rounded for display. The total still matches the exact shoelace area.
Can I use it for a real plot? It is a learning tool on a small grid. For real survey data with bearings and distances, use the traverse calculator.
Is this from the NCERT book? It covers the same topic with fibub’s own plots. fibub is not part of NCERT.
Where the numbers come from
- Heron's formula, the polygon area (shoelace) formula and triangulation by ear clipping are standard mathematics. 1 hectare = 10 000 m² and 1 international acre = 4 046.856 422 4 m² are exact definitions. Written for fibub; the plots are made-up examples.
- Checked against sympy's Polygon.area (a five-sided plot 38.5 square units, a dented plot 24) and on 300 random plots with dents, where the Heron triangles always add up to the shoelace area.
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Last updated 2026-11-09.