Homework 9

The goal of this activity is to mimic Archimedes's computations to find an approximation for the number π. You may use the Pythagorean theorem, coordinate geometry (e.g. the distance formula, since it easily follows from the Pythagoream theorem), and a calculator to approximate the values of square roots to save time (this can be easily done by hand, too, but is just time-consuming). However, you may NOT use trigonometry (for example, you may NOT use a calculator to find sin(22.5^o) - this was not yet available at the time of Archimedes!)

Consider a circle of radius 1. Its circumference is 2π.

• Use squares to obtain a lower bound and an upper bound for the number π as follows.
  • Inscribe a square into the circle. Find its perimeter. Let's denote this perimeter p₁.
  • Circumscribe a square. Find its perimeter. Let's denote this perimeter p₂.
  • Notice that p₁ < 2π < p₂. Divide all parts by 2. What inequality do you obtain for π?
• Repeat the above procedure with regular octagons. You will obtain another inequality.
• Repeat the above procedure with regular 16-gons. You will obtain one more inequality.
  Notice that the more sides, the harder and longer calculations, but the better the estimate.

Remark: Archimedes used polygons with up to 96 sides.