Have you ever wondered how far can you see to the horizon from an elevated position ? Using simple trigonometry the distance to the horizon along the Earth’s surface can be easily determined. Two cases are considered: (i) light travels in a straight line and (ii) light travels along a curved path as a result of atmospheric refraction. In both cases it will be assumed that the Earth is a sphere resulting in a circular cross section when examining the problem in two dimensions.

\(\pi\) is a mathematical constant defined as the ratio of the circumference of a circle to its diameter. There are many other definitions including the ratio of the area of a circle to the square of its radius. It is a constant that also appears in many formulae used in mathematics, physics and engineering.

\(\pi\) is an irrational number. It cannot be expressed as a ratio of two whole numbers $a/b$. Throughout history ingenious ways have been devised to calculate this constant with increasing accuracy. In this article I’ll describe three methods to determine a decimal representation of \(\pi\) (3.14159…):

• The Monte Carlo method where we use a statistical approach to estimate the area of a circle
• The Leibniz formula for \(\pi\) consisting of the evaluation of a simple series
• Machin’s formula for \(\pi\). A more advanced method using a trigonometric relationship