Triangle Area from Coordinates (Shoelace)
Instead of finding side lengths you cross-multiply the coordinates and add — hence “shoelace”. The absolute value means the order of the points does not matter.
Your numbers
Area
19.5
Perimeter
20.4067
Inradius
1.9111
Formula
Area = |x₁ × (y₂ − y₃) + x₂ × (y₃ − y₁) + x₃ × (y₁ − y₂)| ÷ 2
Collinear points give an area of zero, which makes this a quick test of whether three points really form a triangle.
What to enter
| Input | Default | Accepted range |
|---|---|---|
| x₁The horizontal coordinate of the first point. | 1 | No limit |
| y₁The vertical coordinate of the first point. | 1 | No limit |
| x₂The horizontal coordinate of the second point. | 7 | No limit |
| y₂The vertical coordinate of the second point. | 2 | No limit |
| x₃The horizontal coordinate of the third point. | 4 | No limit |
| y₃The vertical coordinate of the third point. | 8 | No limit |
Step by step
Quick reference table
Results when only x₂ changes and everything else stays put.
| x₂ | Area | Perimeter | Inradius |
|---|---|---|---|
| 3.5 | 7.25 | 16.3292 | 0.888 |
| 5.25 | 13.375 | 18.1107 | 1.477 |
| 7 | 19.5 | 20.4067 | 1.9111 |
| 10.5 | 31.75 | 26.0142 | 2.441 |
| 14 | 44 | 32.3161 | 2.7231 |
What each result means
| Result | At default values |
|---|---|
| AreaThe space a shape covers — length squared. | 19.5 |
| PerimeterThe total length around the outside of a shape. | 20.4067 |
| InradiusRadius of the circle touching all sides, sitting inside the shape. | 1.9111 |
Common mistakes
Collinear points give an area of zero, which makes this a quick test of whether three points really form a triangle.
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.
- Area
- The space a shape covers — length squared.
- Perimeter
- The total length around the outside of a shape.
- Inradius
- Radius of the circle touching all sides, sitting inside the shape.
Frequently asked questions
Q. How is Triangle Area from Coordinates (Shoelace) calculated?
Area = |x₁ × (y₂ − y₃) + x₂ × (y₃ − y₁) + x₃ × (y₁ − y₂)| ÷ 2 — Instead of finding side lengths you cross-multiply the coordinates and add — hence “shoelace”. The absolute value means the order of the points does not matter.
Q. Can you walk through an example?
With x₁ 1, y₁ 1, x₂ 7, y₂ 2, x₃ 4, y₃ 8, the answer is Area 19.5.
Q. What do I need to enter?
Enter x₁, y₁, x₂, y₂, x₃, y₃. 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 x₂ moves the answer to x₂ 3.5 → Area 7.25 and x₂ 14 → Area 44. 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?
Collinear points give an area of zero, which makes this a quick test of whether three points really form a triangle.
Related calculators
Keep your length units consistent across inputs — mixing centimetres and metres throws results far off.