Let p(x,y) be a point located arbitrarily in a triangular element. Nodes 1, 2, and 3 are located at the vertices of the triangle and numbered in a counter-clockwise order.

The point p divides the triangle into three subtriangles with areas A₁, A₂, and A₃ opposite each node. An area coordinate, x , is defined as the ratio of the area of a subtriangle to the total area of the element.

The area of any triangle can be found by computing the determinate of a matrix containing the coordinate locations of the triangle vertices. Thus for A₁

And

Collecting terms gives

Here

Thus 2A₁ is a linear function of x and y, and the area coordinate becomes

Likewise

Area coordinates, linear functions of x and y, take on values between 0 and 1 as the point p moves within the element. For example, since x (x,y) is a function of x and y, when the point is located at node 1, x = x₁, y = y₁ and A₁ = 1, A₂ = A₃ = 0. Thus

The properties discussed above are very useful in developing shape functions and formulating stiffness matrices for triangular elements. The Constant Strain Triangle is used as an illustration.

In cartesian coordinates the displacement functions for the CST can be written as:

Here the a’s are determined from the boundary conditions on displacement at the nodes.

The element displacement functions can also be formulated in terms of the area coordinates as follows. For the CST let

Apply the displacement boundary conditions to u(x,y).

Thus

We see that the formulation with area coordinates makes determination of the unknown constants particularly easy. The displacement function for u(x,y) becomes

And for v(x,y)

Since it is usual to order the element node displacement vector in the sequence u₁, v₁, u₂, …, the CST shape function becomes.

Where

We now use the strain-displacement relations to find expressions for the strains within the element. We use the chain rule of differentiation since the displacement functions are indirect functions of x and y.

Where

Thus

and

The extensional strain in the x direction becomes

Where

By the same kind of process we obtain

And

Collecting terms we obtain

The matrix B is

The CST stiffness matrix is found from

And the result is the same as that obtained by using cartesian coordinates in the formulation.