Area of a Triangle by formula (Coordinate Geometry)
Given the coordinates of the three vertices of a triangle ABC, the area can be foiund by the formula below.
Try this
Drag any point A,B,C. The area of the triangle ABC is continuously recalculated using the above formula.
You can also drag the origin point at (0,0).
Given the coordinates of the three vertices of any triangle, the area of the triangle is given by:
where A_{x} and A_{y} are the x and y coordinates of the point A etc..
This formula allows you to calculate the area of a triangle when you know the coordinates of all three
vertices.
It does not matter which points are labelled A,B or C,
and it will work with any triangle, including those where some or all coordinates are negative.
Looking at the formula above, you will see it is enclosed by two vertical bars like this:
The two vertical bars mean "absolute value". This means that it is always positive even if the formula produced a negative result.
Polygons can never have a negative area.
The 'handedness' of point B
If you perform this calculation but omit the last step where you take the absolute value, the result can be negative.
If it is negative, it means that the 2nd point (B) is to the left of the line segment AC.
Here, we mean 'left' in the sense that if you were to stand on point A looking at C, then B is on your left.
If the area is zero
If the area comes out to be zero, it means the three points are
collinear. They lie in a straight line and
do not form a triangle. You can drag the points above to create this condition.
You can also use Heron's Formula
Heron's Formula allows you to calculate the area of a triangle if you know the length of all three sides.
(See Heron's Formula).
In coordinate geometry we can find the distance between any two points
if we know their coordinates,
and so we can find the lengths of the three sides of the triangle, then plug them into Heron's Formula
to find the area.
If one side is vertical or horizontal
In the triangle on the right, the side AC is
vertical (parallel to the y axis).
In this case it is easy to use the traditional "half base times height" method.
See Area of a triangle  conventional method.
Here, AC is chosen as the base and has a length of
8, found by subtracting the y coordinates of A and C. Similarly the altitude is 11, found by subtracting the xcoordinates of B and A.
So the area is half of 8 times 11, or 44.
The box method
You can also use the box method, which actually works for any polygon. For more on this see
Area of a triangle  box method (Coordinate Geometry)
Things to try

In the diagram at the top of the page, Drag the points A, B or C around and notice how the area calculation uses the coordinates.
Try points that are negative in x and y. You can drag the origin point to move the axes.

Click "hide details". Drag the triangle to some random new shape. Calculate its area and then click
"show details" to see if you got it right.

After the above, estimate the area by counting the grid squares inside the triangle. (Each square is 5 by 5 so
has an area of 25).
Once you have done the above, you can click on "print" and it will print the diagram exactly as you set it.
Limitations
In the interest of clarity in the applet above, the coordinates are rounded off to integers and the lengths rounded to one decimal place.
This can cause calculatioons to be slightly off.
For more see
Teaching Notes
While you are here..
... I have a small favor to ask. Over the years we have used advertising to support the site so it can remain free for everyone.
However, advertising revenue is falling and I have always hated the ads. So, would you go to Patreon and become a patron of the site?
When we reach the goal I will remove all advertising from the site.
It only takes a minute and any amount would be greatly appreciated.
Thank you for considering it! – John Page
Become a patron of the site at patreon.com/mathopenref
Other Coordinate Geometry topics
(C) 2011 Copyright Math Open Reference. All rights reserved
