Homework 12

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

• Use regular hexagons to obtain a lower bound and an upper bound for the number Pi as follows.
  • Inscribe a regular hexagon into the circle. Find its perimeter. (Hint: divide the hexagon into 6 equilateral triangles.) Let's denote this perimeter p₁.
  • Circumscribe a regular hexagon. Find its perimeter. (Hint: divide the hexagon into 6 equilateral triangles. Use the Pythagorean Theorem to find the length of the sides of these triangles.) Let's denote this perimeter p₂.
  • Notice that p₁ < 2Pi < p₂. What inequality do you obtain for Pi?
• 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.