# Area of Triangle in Coordinate Geometry: Formula & Examples

## TL;DR
The area of a triangle in coordinate geometry is found from its three vertices with the formula Area=12∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣, the shoelace formula. This article derives the formula, works through six examples, and shows how a result of zero proves the three points are collinear.

## What Is The Area Of A Triangle In Coordinate Geometry?
The **area of a triangle in coordinate geometry** is the amount of flat space a triangle covers when its three corners are given as points (x1,y1), (x2,y2), and (x3,y3) on the coordinate plane. Instead of measuring a base and a perpendicular height, you compute the area straight from those six numbers using one formula. The result is always a positive value, expressed in square units.

This matters because in most real problems you know **where** the corners sit, not how tall the triangle is. The coordinate approach turns a measuring job into an arithmetic one. It sits inside the wider subject of [coordinate geometry](/content/math/geometry/coordinate-geometry/index.html).

**For Example:** Surveyors mapping a plot of land never measure the height of a triangle directly; they read three corner coordinates off a GPS and let the formula do the rest.

## The Area Formula And What Each Term Means
For a triangle with vertices A(x1,y1), B(x2,y2), and C(x3,y3), the area is:

Area=12∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣

Reading the formula piece by piece:
- Each x-coordinate is multiplied by the **difference of the other two vertices' y-coordinates**.
- The three products are added together.
- The absolute value strips any minus sign, because an area cannot be negative.
- The 1/2 scales the result down.

### The Determinant Form
The same formula can be written as a 3×3 **determinant**:
Area=12∣det⁡[x1y11x2y21x3y31]∣

Expanding this determinant reproduces the shoelace expression above.

## Where The Formula Comes From
The shoelace formula is not something to memorise blindly; it drops out of a picture. It is derived from the areas of trapeziums formed under the triangle’s sides.

## Examples Of Area Of A Triangle In Coordinate Geometry

### Example 1
**Find the area of the triangle with vertices A(1,1), B(4,1), and C(1,5).**

Substitute:  
Area=12∣1(1−5)+4(5−1)+1(1−1)∣=12∣−4+16+0∣=6 square units

### Example 2
**Find the area of the triangle with vertices A(3,4), B(4,7), and C(6,−3).**  
Area=12∣3(7−(−3))+4((−3)−4)+6(4−7)∣=8 square units

### Example 3
**Verify whether A(1,5), B(2,3), and C(−2,−11) are collinear.**  
Area=12∣1(3−(−11))+2((−11)−5)+(−2)(5−3)∣=11 square units

### Example 4
**The vertices of a triangle are A(0,0), B(a,0), and C(0,b). Find its area in terms of a and b.**  
Area=12∣ab∣=12ab square units

### Example 5
**Find the area of a triangular park with corners A(2,1), B(7,2), and C(4,6).**  
Area=11.5 km²

## Where the Formula Earns Its Keep
- **Land surveying.** A surveyor records boundary corners as coordinates.
- **Computer graphics.** Every 3D model is a mesh of triangles.
- **Navigation and GIS.** Mapping software layers polygons over the world.

## The Mistakes Students Make Most Often
### Mistake 1: Dropping the absolute value

### Mistake 2: Forgetting the one-half

### Mistake 3: Scrambling the vertex order

## Conclusion
- The area formula is 12∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣.
- A result of **zero** proves the three points are collinear.