As I understand it, the Pythagorean Snail starts with an isosceles right-angled triangle with shorter sides of unit length, making the hypotenuse length sqrt(2). This hypotenuse forms a side of the next triangle built on it. The other side is of unit length so the new hypotenuse is sqrt(2+1)=sqrt(3). The next triangle has a hypotenuse of sqrt(4)=2, and so on. The area of the first triangle is (1/2)(1*1)=1/2; the second triangle has the same height but the base is sqrt(2), making its area sqrt(2)/2; the third triangle has area sqrt(3)/2; the fourth sqrt(4)/2=1; and so on. We can write the sum of the areas: (1/2)(1+sqrt(2)+sqrt(3)+...+sqrt(N)) for N triangles. For N=16, the total area is 22.2346 approx.
As for a formula, the best we may be able to come up with is the area under the curve y=sqrt(x)/2.
This is given by the integral(ydx)=(1/2)integral(x^(1/2)dx)=1/3x^(3/2)=N^(3/2)/3=64/3=21.3 when N=16. Compare this with the actual 22.2 we can see that it's not far from the true value (96% accurate).