Given the coordinates of the three vertices of a triangle ABC, the area 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 at the top of the page, you will see it is enclosed by two vertical bars like this:

Note: 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 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

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 x-coordinates 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

1. 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.
2. 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.
3. 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).

Other Coordinate Geometry entries

• Introduction to coordinate geometry
• The coordinate plane
• Coordinates of a point
• Distance between two points

• Introduction to Lines in Coordinate Geometry
• Lines (in Coordinate Geometry)
• Rays (in Coordinate Geometry)
• Line Segments (in Coordinate Geometry)
• Midpoint of a line segment (Midpoint Theorem)
• Slope (m) of a line
• Intercept (b) of a line
• Equation of a line in slope-intercept form y=mx+b
• Equation of a line in point-slope form y=m(x-Px)+Py

• Vertical lines
• Horizontal lines
• Parallel lines
• Perpendicular lines
• Intersecting lines

• Area of a triangle (formula method)
• Area of a triangle (box method)

• Print blank graph paper