The Pythagorean snail or spiral starts with an isosceles right-angled triangle with, in this case, shorter sides of length 2cm, making the hypotenuse length 2sqrt(2)cm. The area of this first triangle is half the area of a 2cm square=2sq cm. The hypotenuse forms a side of the next triangle built on it. The other side is 2cm so the new hypotenuse is 2sqrt(2+1)=2sqrt(3). The next triangle has a hypotenuse of 2sqrt(4)=4, and so on. The second triangle has height 2cm but the base is 2sqrt(2), making its area 2sqrt(2); the third triangle has area 2sqrt(3); the fourth 2sqrt(4)=4; and so on. We can write the sum of the areas: 2(1+sqrt(2)+sqrt(3)+...+sqrt(N)) for N triangles. For N=16, the total area is 88.94 sq cm approx.
As for a formula, the best we may be able to come up with is the area under the curve y=2sqrt(x)/2. This is given by the integral(ydx)=2*integral(x^(1/2)dx)=(4/3)x^(3/2)=4N^(3/2)/3=256/3=85.33 when N=16. Compare this with the actual 88.94 sq cm, we can see that it's not far from the true value (96% accurate).