Area of a Triangle from Coordinates Calculator
This area of a triangle from coordinates calculator works out the area from the three vertices alone, exactly, with no side lengths and nothing rounded.
Type (2, 4) (3, -6) (7, 8) and the answer is 27 square units. Three multiplications got it there, and not a square root in sight.
One Formula, Three Multiplications
Finding the area of a triangle with vertices (x₁, y₁), (x₂, y₂) and (x₃, y₃) takes one line:
Area = ½ |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|
Each x is multiplied by the difference of the other two y values, and the pattern rotates: the first takes second minus third, the second takes third minus first, the third takes first minus second.
Keeping that rotation straight is where almost every mistake happens. Write the three brackets out empty before you put a single number in any of them — the triangle area from 3 points is one addition once they are filled.
| Vertices | Area |
|---|---|
| (2, 4) (3, −6) (7, 8) | 27 |
| (0, 0) (4, 0) (0, 3) | 6 |
| (0, 0) (1, 0) (0, 1) | ½ |
| (1, 1) (2, 2) (3, 3) | 0 — collinear |
Zero Is an Answer, Not a Failure
Look at the last row. Three points, and no area at all.
That happens for exactly one reason: they lie on a single straight line, so there is nothing enclosed. Which makes this calculation the standard way of answering a question that never mentions area — show that these three points are collinear. Work out the area; if it comes to zero, you have shown it.
It is worth knowing that the two questions are the same question, because exam papers ask the second one far more often than the first.
The Sign Everybody Throws Away
Before the absolute value goes on, the result can be negative. That negative is not an error to be tidied up — it is information.
A positive value means the three vertices run anticlockwise in the order you listed them; a negative one means clockwise.
Swap any two of them and the sign flips while the size stays put. That is a neat way to see that the labelling never affected the area in the first place, only the route the working took to reach it.
Try (0, 0), (1, 0), (0, 1) and then the same three points with the last two swapped. Same half-unit of area, opposite signs. No other part of this calculation tells you anything about the shape rather than its size.
Why Bother, When the Side Lengths Would Do
You could. Work out the three side lengths, then apply Heron's formula. Several calculators do exactly that, and at least one well-known one does it as its only method.
It is the long way round, and it costs you the exact answer.
Side lengths come out as square roots almost every time. Multiply four of those together, take another root of the result, and a clean whole number arrives with a trail of digits after it. The determinant never leaves the rational numbers at all.
If you do want the side lengths for their own sake — for a perimeter, say — the distance formula calculator is the page for that, and it keeps its surds exact too.
The Answer Is Always a Whole Number or a Half
Give it whole-number vertices and you will never get a third, or a quarter, or 4.7.
The reason is in the formula itself. Everything inside the bars is a sum of products of integers, so it is an integer, and the only thing that happens afterwards is halving.
So the area lands on a whole number or a half, always — which makes a free check on your own working. Arrive at 8.3 and something went wrong before the halving.
It also means a calculator printing 4.5 and one printing 9/2 are giving the same answer, and the second one has not thrown anything away.
A Worked Example, Start to Finish
Take A(2, 4), B(3, −6), C(7, 8), which is the set most textbooks reach for.
First bracket: 2 times (−6 − 8), which is 2 × −14 = −28. Second: 3 times (8 − 4), so 3 × 4 = 12. Third: 7 times (4 − (−6)), and that minus-minus is where the marks go — it is 7 × 10 = 70, not 7 × −2.
Add them: −28 + 12 + 70 = 54. Halve it: 27. The result was positive, so A, B and C run anticlockwise in that order.
Three multiplications and one addition. No square roots appeared anywhere, and the answer needed no rounding — which is the whole argument for finding a triangle area from 3 points this way rather than through their lengths.
Where the Formula Comes From
The expression inside the bars is a determinant — the one belonging to the matrix built from the three vertices with a column of ones.
Calling it a determinant is not notation dressed up as depth. A 2×2 determinant measures the area of the parallelogram spanned by two vectors, and a triangle is half a parallelogram. Take AB and AC as the two vectors and half their determinant is the area, which is exactly what the formula computes once the brackets are expanded. The determinant calculator handles that object on its own, for matrices of any size.
It also generalises, which is worth a sentence because the generalisation has a name and a job. Walk round any polygon writing down x₁y₂ − x₂y₁ for each edge in turn, add them all up and halve, and you have its area — the same calculation this page does, just carried on past the third vertex. It is called the shoelace formula, after the way the multiplications cross over each other like threading a shoe, and surveyors have used it on plots of land since long before anyone typed coordinates into anything. A triangle is simply the shortest case, where the walk closes after three steps.
What Else Three Points Will Tell You
The same three coordinates answer more than one question, and it is worth knowing which is which.
Triangle area from 3 points needs all three together. The midpoint of any side needs only two of them, and is the natural next step when a question moves on to medians or a centroid. Neither calculation feeds the other — they simply happen to start from the same numbers.
How to Use This Area of a Triangle from Coordinates Calculator
Type all three vertices on one line: (2, 4) (3, -6) (7, 8). Brackets are optional and both commas and spaces separate, so 2,4 3,-6 7,8 works just as well.
An A = in front of a point is ignored, so a line copied straight out of a textbook works. Coordinates may be negative, decimal or fractions. The area comes back exact.
Area of a Triangle from Coordinates Calculator FAQ
What is the formula for the area of a triangle with coordinates?
Why is my area coming out zero?
Does the order of the vertices matter?
Can the area be negative?
Do I need the side lengths first?
Why is the answer a fraction?
Does this work with negative coordinates?
What if two vertices are the same point?
Three multiplications, one addition, halve it, drop the sign. If the result is zero the points are collinear, and if you were given whole numbers the answer will land on a whole number or a half.
