Homework 10

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

• Use regular polygons 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 everything by 2 to obtain an inequality for π: p₁/2 < π < p₂/2.
• 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.