Area of Triangle Using Coordinates of Vertices
In this method, if coordinates of the vertices of a triangle are given then we will see how to found the area of the triangle.
If coordinates of the triangle are (x1, y1), (x2, y2), and (x3, y3) then the area of the triangle is given by:
Derivation of the Formula
Given triangle PQR with coordinates P(x1, y1), Q(x2, y2), and R(x3, y3), lets find the formula for its area:
Step 1: Draw the perpendiculars from coordinates P, Q, and R to X-axis at A, B, and C respectively.
Step 2: Now if we look at the figure carefully, three different trapeziums are formed such as PQAB, PBCR, and QACR in the coordinate plane.
Step 3: So the area of ∆QPR is calculated as
Area of ∆PQR = [Area of trapezium PQAB + Area of trapezium PBCR] – [Area of trapezium QACR] . . . (1)
Step 4: Now calculating areas of all 3 trapeziums.
Since Area of a trapezium = (1 / 2) (sum of the parallel sides) × (distance between sides)
Finding Area of a Trapezium PQAB
⇒ Area of trapezium PQAB = (1 / 2)(QA + PB) × AB
⇒ QA = y2
⇒ PB = y1
⇒ AB = OB – OA = x1 – x2
⇒ Area of trapezium PQAB = (1 / 2)(y1 + y2)(x1 – x2 ) . . . (2)
Finding Area of a Trapezium PBCR
⇒ Area of trapezium PBCR =(1 / 2) (PB + CR) × BC
⇒ PB = y1
⇒ CR = y3
⇒ BC = OC – OB = x3 – x1
⇒ Area of trapezium PBCR =(1 / 2) (y1 + y3 )(x3 – x1) . . . (3)
Finding Area of a Trapezium QACR
⇒ Area of trapezium QACR = (1 / 2) (QA + CR) × AC
⇒ QA = y2
⇒ CR = y3
⇒ AC = OC – OA = x3 – x2
⇒ Area of trapezium QACR =(1 / 2)(y2 + y3 ) (x3 – x2 ). . . (4)
Step 5: Substituting (2), (3) and (4) in (1),
⇒ Area of ∆PQR = (1 / 2)[(y1 + y2)(x1 – x2 ) + (y1 + y3 )(x3 – x1) – (y2 + y3 ) (x3 – x2 )]
⇒ Area of ∆PQR = (1 / 2) |[x1 (y2 – y3 ) + x2 (y3 – y1 ) + x3(y1 – y2)]|
Therefore, this is the formula to find the area of triangle if coordinates are given.
Note: Observe that there is a mod, which indicates that, if we got a negative value we should only consider the numerical value as the area can’t be negative.
Area of a Triangle in Coordinate Geometry
Coordinate geometry is defined as the study of geometry using the coordinate points on the plane with any dimension. Using coordinate geometry, it is possible to find the distance between two points, divide lines in a ratio, find the mid-point of a line, calculate the area of a triangle in the Cartesian plane etc.
There are various methods to find the area of the triangle according to the parameters given, like the base and height of the triangle, coordinates of vertices, length of sides, etc. In this article, we will discuss the method of finding area of any triangle when its coordinates are given.