Quadrea

Quadrea (𝒜) is the measure of size of a triangle. It is also equal to the 16 times the area squared.

Heron's/Archimedes's
In terms of the quadrance of the sides, the quadrea (𝑄) is

$$ 16A^2 = 4Q_1Q_2-(Q_1+Q_2-Q_3)^2 = (Q_1+Q_2+Q_3)^2-2(Q_1^2+Q_2^2+Q_3^2) $$

Spread
If the quadrea is known, the spreads of the triangle are simple

Addings
Adding quadreas is a matter of solving the following formula, assuming your know $$x_1$$ and $$x_2$$ and are looking for their sum, $$x_3$$:

$$(x_3-x_1-x_2)^2 = 4x_1x_2$$

While area is a fairly intuitive concept, it can be hard to work with at times. Consider Heron's formula, which says the area of a triangle is equal $$\sqrt{s(s-a)(s-b)(s-c)}$$, where a, b, and c are the side lengths, and s the semi-perimeter, i.e. $$\frac{a+b+c}{2}$$. This might generate four denominators of one-half per term, and be a root. It makes much more sense to deal in rational quantities, and only attempt the infinite process of square-rooting at the very end, rather than along the way.