How to Find the Area of a Triangle From the Coordinates of Its Vertices

Geometry Lessons

The short answer

A = ½|x1(y2-y3) + x2(y3-y1) + x3(y1-y2)|

Each x is multiplied by the gap between the other two y values. Add the three products, take the size of the total, then halve it. The vertical bars mean ignore a minus sign if you get one.

This is the method for when you have coordinates and nothing else. No side lengths measured, no angles taken, no perpendicular you could get at. It is called the shoelace formula because of the criss cross pattern the multiplications make when you write the coordinates out in a column, and it works on any polygon, not just triangles.

Why it works

The formula is really about signed areas. Take each edge of the triangle and imagine the trapezoid dropped from it down to the x axis. Walk around the triangle and you add some of those trapezoids and subtract others, and everything outside the shape cancels out exactly, leaving the area inside.

That cancelling is also where the sign comes from. Travel around the vertices counterclockwise and the total comes out positive. Travel clockwise and you get the same number with a minus in front. The area itself has not changed, only the direction you walked, which is why you take the size at the end and stop worrying about it.

1

Label the points

Call them (x1,y1), (x2,y2) and (x3,y3). Any order works, as long as you keep it consistent.

2

Cross multiply

Each x multiplied by the difference of the other two y values, always in the same rotation.

3

Add, take the size, halve

Sum the three products, drop any minus sign, then divide by two.

A worked example

Take vertices at (1,1), (5,2) and (3,6). The first product is 1 times the gap between 2 and 6, which is 1 times negative 4, or negative 4. The second is 5 times the gap between 6 and 1, which is 5 times 5, or 25. The third is 3 times the gap between 1 and 2, which is 3 times negative 1, or negative 3.

Add them and you get negative 4 plus 25 plus negative 3, which is 18. Halve that and the area is 9 square units. If you had listed the same three points in the other rotation the total would have been negative 18, and halving its size still gives 9.

A triangle plotted on a grid with vertices at 1 comma 1, 5 comma 2 and 3 comma 6, beside the shoelace working showing products of minus 4, 25 and minus 3 summing to 18 and halving to an area of 9.
Two of the three products come out negative here, which is normal and not a sign of an error. Only the total matters, and only its size.

A minus sign is a direction, not an error

The most common worry with this method is a negative result, and it is not a mistake. It only records which way round you listed the corners. The vertical bars in the formula are there precisely to throw that information away once you have finished.

counterclockwisegives(-4) + 25 + (-3)9

a positive total, halved as it stands

clockwisegives4 + (-25) + 39

the same size of total, so the same area once you drop the sign

When three points do not make a triangle

If the three points happen to sit on a straight line, the products cancel completely and the total comes out as zero. That is the formula telling you there is no area to find, which is correct rather than a failure. Points at (1,1), (2,3) and (3,5) all sit on the same line, and the method returns zero for them.

Where this actually gets used

Coordinates are how land parcels, site plans and CAD drawings store their geometry, so this is often the only method available. A surveyor with corner coordinates from a total station does not have a base and a height, and measuring one off a drawing would introduce error the coordinates do not have. The shoelace method uses the numbers exactly as recorded.

It also scales in a way the other formulas do not. A triangular lot and a lot with nine corners are the same problem, which is why this is the method that ends up inside almost every mapping and drafting package.

Mistakes that catch people out

  • Treating a negative total as an error. It records the rotation you used. Take the size and move on.
  • Forgetting to halve. The sum of the products is twice the area, not the area.
  • Mixing up the rotation partway through. Go around the corners in one consistent direction from start to finish.
  • Subtracting y values in the wrong order. The pattern is each x times the y after it minus the y before it. Reversing one term quietly breaks the total.
  • Rounding coordinates before you start. Small changes in the corners move the area more than you would expect, so keep the decimals until the end.

Practice questions

  1. Vertices at (0,0), (4,0) and (0,3). Find the area.
  2. Vertices at (0,0), (6,0) and (0,8). Find the area.
  3. Vertices at (2,1), (6,1) and (4,5). Find the area.
  4. Vertices at (1,1), (2,3) and (3,5). Find the area.
  5. Vertices at (0,0), (5,1) and (2,6). Find the area.
Show the answers
  1. ½|0(0-3) + 4(3-0) + 0(0-0)| = ½|12| = 6 square units
  2. ½|0(0-8) + 6(8-0) + 0(0-0)| = ½|48| = 24 square units
  3. ½|2(1-5) + 6(5-1) + 4(1-1)| = ½|-8 + 24 + 0| = 8 square units
  4. the total is zero, so the three points sit on a straight line and enclose no area
  5. ½|0(1-6) + 5(6-0) + 2(0-1)| = ½|30 – 2| = 14 square units

Common questions

What is the shoelace formula?

It finds the area of a polygon from the coordinates of its corners. For a triangle, multiply each x by the difference of the other two y values, add the three products, take the size of the total and halve it.

Why did I get a negative area?

You listed the vertices clockwise rather than counterclockwise. The size of the number is still correct, so drop the minus sign and carry on.

Does the order of the vertices matter?

Only the rotation matters, not which corner you start from. Going around consistently in either direction gives the right answer once you take the size.

Can I use it on shapes with more than three sides?

Yes. The same method extends to any polygon with straight edges, as long as you go around the corners in order and the edges do not cross.

Do I need the side lengths?

No. That is the point of the method. It works from the coordinates alone, with no distance calculated and no angle measured.

What if all three points lie on one line?

The total comes out as zero, which means there is no triangle. Three collinear points enclose no area, and the formula reports that correctly.

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