How to compute the area on top of this red green line?

Question

https://en.wikipedia.org/wiki/Gini_coefficient

I says that if u people own f portion of the wealth, then the gini index is f-u.

They draw some triangle there

It seems that the area on top of the green line and to the left of the red line is simply .5(f-u)

However, I know no easy way to compute that. Pencil pushing confirm the calculation.

Are there simpler ways?

I know the formula hold if f is equal to 1. When f is less than 1 the area of the triangle shrink proportionally. Well, not really. So yea I am looking for better derivation here.

Note: f + u is NOT 1

It's u people have f incomes.

Answer 1

The triangle of interest has coordinates $(0,0)$, $(1-u,1-f)$ and $(1,1)$ when visited in counterclockwise order. So compute its area as a determinant using the shoelace formula: $$ \frac12\left| \begin{matrix} 1&1&1\\ 0&1-u&1\\ 0&1-f&1\\ \end{matrix} \right|=\frac12\left((1-u) - (1-f)\right)=\frac12(f-u). $$ Using the shoelace formula you can prove that a triangle with one vertex at the origin and the others at $(a,b)$ and $(c,d)$ will have area $\frac12|ad-bc|$.

Answer 2

First of all, note that $$ f+u = 1 $$ is assumed.

Now you will notice that the area $A$ can be calculated as $(A+B+C+D) - B - C - D$.

1. $A+B+C+D$

It is nothing but a right triangle with width 1 and height 1.

Therefore, $$ A+B+C+D = \frac{1}{2}(1 \times 1) = 0.5 $$

2. $B$ and $C$

It is a right triangle with width $u$ and height $f$.

$$ B = C = \frac{1}{2}(f \times u) $$

3. $D$

It is square, therefore

$$ D = u^2 $$

Now we have

$$ A = \frac{1}{2} - fu - u^2 $$ $$ = \frac{1}{2} - u(f + u) $$ , and $$ = \frac{1}{2} - u $$ because $f+u =1$.

However, from $1=f+u$, $$ \frac{1}{2} = \frac{1}{2}(f+u) $$ , and we have However, from $1=f+u$, $$ \frac{1}{2} - u = \frac{1}{2}(f-u) $$

Answer 3

Why didn't I notice this.

Take a line from top left of square D to the left hitting the hypotenuse. The length of that line is f-u. Let's called that segment e

e divides triangle A into 2 regions. e, is a base of those 2 triangles. The sum of their height is 1.

Hence, the area of A is simply .5(f-u)*1 which is what we want.

How do I modify the pictures and write math notation better? What software you use?