Quadrilateral Area from Coordinates
This is the shoelace formula. List the corners in the order you walk around them, cross-multiply the x and y values, and halve the difference. It holds for shapes that are not rectangles and even for ones with a dent.
Your numbers
Area
63
Perimeter
32.2543
Diagonal 1
12.0416
Diagonal 2
10.8167
Formula
Area = |x₁y₂ − x₂y₁ + x₂y₃ − x₃y₂ + x₃y₄ − x₄y₃ + x₄y₁ − x₁y₄| ÷ 2
The corners must go around the shape, clockwise or anticlockwise. Listing them across a diagonal gives too small an area.
What to enter
| Input | Default | Accepted range |
|---|---|---|
| x₁The horizontal coordinate of the first point. | 2 | No limit |
| y₁The vertical coordinate of the first point. | 1 | No limit |
| x₂The horizontal coordinate of the second point. | 12 | No limit |
| y₂The vertical coordinate of the second point. | 2 | No limit |
| x₃The horizontal coordinate of the third point. | 11 | No limit |
| y₃The vertical coordinate of the third point. | 9 | No limit |
| x₄The horizontal coordinate of the fourth point. | 3 | No limit |
| y₄The vertical coordinate of the fourth point. | 8 | No limit |
Step by step
Quick reference table
Results when only x₁ changes and everything else stays put.
| x₁ | Area | Perimeter | Diagonal 1 |
|---|---|---|---|
| 1 | 66 | 33.4588 | 12.8062 |
| 1.5 | 64.5 | 32.8397 | 12.4197 |
| 2 | 63 | 32.2543 | 12.0416 |
| 3 | 60 | 31.1887 | 11.3137 |
| 4 | 57 | 30.2667 | 10.6301 |
What each result means
| Result | At default values |
|---|---|
| AreaThe space a shape covers — length squared. | 63 |
| PerimeterThe total length around the outside of a shape. | 32.2543 |
| Diagonal 1The length of the first diagonal. | 12.0416 |
| Diagonal 2The length of the second diagonal. | 10.8167 |
Common mistakes
The corners must go around the shape, clockwise or anticlockwise. Listing them across a diagonal gives too small an area.
Glossary
- x₁
- The horizontal coordinate of the first point.
- y₁
- The vertical coordinate of the first point.
- x₂
- The horizontal coordinate of the second point.
- y₂
- The vertical coordinate of the second point.
- x₃
- The horizontal coordinate of the third point.
- y₃
- The vertical coordinate of the third point.
- x₄
- The horizontal coordinate of the fourth point.
- y₄
- The vertical coordinate of the fourth point.
- Area
- The space a shape covers — length squared.
- Perimeter
- The total length around the outside of a shape.
- Diagonal 1
- The length of the first diagonal.
- Diagonal 2
- The length of the second diagonal.
Frequently asked questions
Q. How is Quadrilateral Area from Coordinates calculated?
Area = |x₁y₂ − x₂y₁ + x₂y₃ − x₃y₂ + x₃y₄ − x₄y₃ + x₄y₁ − x₁y₄| ÷ 2 — This is the shoelace formula. List the corners in the order you walk around them, cross-multiply the x and y values, and halve the difference. It holds for shapes that are not rectangles and even for ones with a dent.
Q. Can you walk through an example?
With x₁ 2, y₁ 1, x₂ 12, y₂ 2, x₃ 11, y₃ 9, x₄ 3, y₄ 8, the answer is Area 63.
Q. What do I need to enter?
Enter x₁, y₁, x₂, y₂, x₃, y₃, x₄, y₄. The result recalculates as you type, and an empty box counts as zero.
Q. How much does the answer move if I change a number?
Changing only x₁ moves the answer to x₁ 1 → Area 66 and x₁ 4 → Area 57. The table below lays out five steps.
Q. How are the numbers rounded?
Money is shown to the nearest whole unit, percentages to one decimal place and everything else to two. What you see is rounded; the calculation itself carries the unrounded value forward.
Q. Anything to watch out for?
The corners must go around the shape, clockwise or anticlockwise. Listing them across a diagonal gives too small an area.
Related calculators
Keep your length units consistent across inputs — mixing centimetres and metres throws results far off.