Library / Geometry, coordinates, surveying and trigonometry / GEO-28
Geometry, coordinates, surveying and trigonometry
Triangle area from Cartesian coordinates
Area of a triangle from the coordinates of its three corners in one frame.
- Ref
- GEO-28
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- A [cm²]
For corners at (x₁, y₁), (x₂, y₂) and (x₃, y₃) the area is half the absolute value of the cross product of two edge vectors from the first corner. The absolute value makes the result independent of the order of the corners. If the three points lie on one line the value is zero. This is the three-point case of the shoelace formula.
Area of a triangle from the coordinates of its three corners in one frame. The absolute value removes the dependence on the order of the vertices, and collinear points give zero.
Calculator
- Signed value of the cross product, 2A with sign12 cm²
- Side lengths: perimeter11.4 cm
- Height above the side 1–23 cm
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The triangle with corners (1, 2), (5, 2) and (2, 5) has an area of 6 cm².
- Moving one corner along a line parallel to the opposite side leaves the area unchanged, since the base and height stay the same.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (cm²) | Local step | Change in result |
|---|---|---|---|---|---|---|
| y1 Vertex 1, y | −10 to 10 cm | 0 | 24 | 24 | +0.4 cm | −10 % |
| y3 Vertex 3, y | −10 to 10 cm | 0 | 24 | 24 | +0.4 cm | +13.3 % |
| x1 Vertex 1, x | −10 to 10 cm | 0.5 | 22.5 | 22 | +0.4 cm | −10 % |
| x2 Vertex 2, x | −10 to 10 cm | 0.5 | 16.5 | 16 | +0.4 cm | +10 % |
| y2 Vertex 2, y | −10 to 10 cm | 2 | 12 | 10 | +0.4 cm | −3.33 % |
| x3 Vertex 3, x | −10 to 10 cm | 6 | 6 | 0 | +0.4 cm | 0 % |
Sweep of Vertex 3, y
| Vertex 3, y (cm) | Triangle area (cm²) |
|---|---|
| −10 | 24 |
| −7.5 | 19 |
| −5 | 14 |
| −2.5 | 9 |
| 0 | 4 |
| 2.5 | 1 |
| 5 | 6 |
| 7.5 | 11 |
| 10 | 16 |
Sweep one variable
Variables and how each one contributes
Triangle area
What the equation gives you. Everything else feeds it.
Vertex 1, x
The x-coordinate of vertex 1, in cm. Changes the area in proportion to the difference between the other two heights. Raising it by 0.5 (to 1.5) moves the result by −12.5 %, from 6 to 5.25 cm².
Vertex 1, y
The y-coordinate of vertex 1, in cm. Changes the area in proportion to the difference between the other two widths. Raising it by 0.5 (to 2.5) moves the result by −12.5 %, from 6 to 5.25 cm².
Vertex 2, x
The x-coordinate of vertex 2, in cm. Changes the area in proportion to the difference between the other two heights. Raising it by 0.5 (to 5.5) moves the result by +12.5 %, from 6 to 6.75 cm².
Vertex 2, y
The y-coordinate of vertex 2, in cm. Changes the area in proportion to the difference between the other two widths. Raising it by 0.5 (to 2.5) moves the result by −4.17 %, from 6 to 5.75 cm².
Vertex 3, x
The x-coordinate of vertex 3, in cm. Changes the area in proportion to the difference between the other two heights. Raising it by 0.5 (to 2.5) moves the result by +0 %, from 6 to 6 cm².
Vertex 3, y
The y-coordinate of vertex 3, in cm. Changes the area in proportion to the difference between the other two widths. Raising it by 0.5 (to 5.5) moves the result by +16.7 %, from 6 to 7 cm².
Worked example with the values above
- Edge vectors from vertex 1: (4, 0) and (1, 3).
- Cross product: 4 × 3 − 1 × 0 = 12.
- A = |12| / 2 = 6 cm².
Assumptions and limits
- Plane (two-dimensional) Cartesian coordinates in one consistent unit.
- The two points or the two direction vectors are not the same, and neither vector has zero length.
Common mistakes
- Forgetting the absolute value or the factor of one half.
- Mixing the order of the subtracted products in the cross product.
- Using vertices from different frames or in different units.
Statistics and uncertainty
These are exact algebraic results from the coordinates, so the uncertainty comes from the inputs. Near-parallel lines and nearly collinear points are the cases where small input errors give large changes in angles or areas.
Where it is used
- Clearances, offsets and angles between members in drawings, CAD and surveying.
- Areas of plots, plates and polygon pieces from corner coordinates.
Maths you need
- Gradient: rise over run of a line
- Vector: direction and length written as (x, y)
- Dot product: the sum of products of components