# Nerding out at the trailhead

Well there we were, at the trailhead for our backpacking trip into Altos de Lircay when a student checked the date. It was Pi day! March 14th: 3-14: π! The day the circle was invented! One of the most significant days in mathematics, second only to e-day (e=2.71828, so e-day would be the 72nd of February.) Our group decided to celebrate the occasion by taking a close look at how pi is calculated. Pi is the ratio between the circumference(perimeter) of a circle and the diameter, and a circle can be viewed as a regular polygon with infinite sides.

Unfortunately, we were on a backpacking trip rather than at desks with a whiteboard and calculators, so am improvisation was necessary. Luckily, we had to drive about 70km on dirt roads to get where we were and the rental car was caked in dust, making an ideal etch-a-sketch.

We started off specific and determined the perimeter of an octagon measuring 10 inches across. McKenna saw that we could divide the shape up into 8 triangles like a pizza and Brady had just learned the Law of Sines in geometry, so we soon had an expression for the perimeter of the octagon. Burke and Isabelle further refined the expression by applying exact values to trig functions and the half-angle identity they’ve been studying throughout the last unit.

Our algebra students went on to make our perimeter equation more and more general, first by replacing the 10in distance with a d and then using n to represent the number of sides. Now we have an equation for the perimeter of any right polygon, and a rather complex expression for the ratio between perimeter and distance across as a function of the number of sides. To finish up, McKenna applied the limit process of calculus to give our polygon infinite sides. Aka: a circle. Pie. Pi.