The classical Newton iteration formula to find a root of the equation \(f(x)= 0\) where \(f\) is a sufficiently differentiable real function is \(x_{n+1}= x_n- f(x_n)/f'(x_n)\). Because \[ f(x)= f(x_n)+ \int^x_{x_n} f'(y)\,dy, \] the Newton iteration formula can be interpreted as an approximation of the integral by \((x- x_n)\cdot f'(x_n)\). If we use the trapezoidal approximation, we obtain an \textit{S. Weerakoon} and \textit{T. G. I. Fernando} arithmetic mean method [Appl. Math. Lett. 13, No. 8, 87--93 (2000; Zbl 0973.65037)]. The harmonic mean method variant was considered by \textit{A. Y. Özban} [Appl. Math. Lett. 17, No. 6, 677--682 (2004; Zbl 1065.65067)]. The geometrical mean leads to the formula \[ x_{n+1}= x_n- {f(x_n)\over \text{sign}(f'(x_0))\cdot \sqrt{f'(x_n)\cdot f'(v_{n+1})}}, \] where \(v_{n+1}= {f(x_n)\over f'(x_n)}\). It is shown that the order of convergence of the geometrical mean method is cubical for a simple root and linear for a multiple root. The values of the corresponding asymptotic error constant \(\lim_{x\to\infty} {x_{n+1}\over (x_n-\alpha)^p}\), where \(\alpha\) is the root and \(p\) the order of convergence are determined. A comparison of the efficiency of the geometrical mean method with other mean methods is also included.