
“real: chapter_03” — 2011/5/22 — 22:50 — page 19 — #19
Sequences and Series 3-19
This representation of π as a limit serves actually as a basis for numer-
ical computations which can be explained as follows. Using Figure
3.1 for reference, we construct lines touching the circle and parallel
to the sides of the inscribed n-gon. These lines form a circumscribed
polygon similar to the inscribed n-gon and having larger dimensions.
Indeed if b and B are, respectively, the sides opposite to the angle
2π/n in the above triangles and h, H are their respective heights
then we have (from elementary geometry), H = 1, h = cos π/n,
h/H = b/B, f
n
= (n/2) bh, F
n
= (