In this paper the length of the arc of a circle and the area of a
circular sector are defined in an elementary way. From this we
derive the geometrical definition of the number $\pi$, in two
equivalent ways. Formulas for the area and the perimeter of a
circle are proved. Also, the number $\pi$ is represented as a
limit of several sequences involving nested radicals. From this
some approximations of $\pi$ are obtained. One of them is in
terms of Golden ratio.