Homework 10

Approximate the number pi as follows. Consider a circle of radius 1. Its circumference is 2Pi.

Part 1 (Did in class on 4/21).

• 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 procedure described in Part 1 with regular octagons and with regular 16-gons. You will obtain two new inequalities. Notice that the more sides, the harder and longer calculations, but the better the estimate.

Remark: Archimedes used polygons with up to 96 sides.