Correct option is A
Given:
On a large flat surface,
identical circular coins are placed in the
most compact arrangement without overlapping. You are given
unlimited coins, and asked to find the
fraction of surface area not covered.
Concept:
Circle Packing in Two Dimensions: Hexagonal Packing: The most efficient arrangement of circles (coins) on a plane is the hexagonal packing arrangement. In this arrangement, each circle is surrounded by six others, creating a repeating pattern similar to a honeycomb.
Packing Density: Formula for Packing Density: In a hexagonal packing, the highest packing density, i.e., the fraction of the area covered by the coins, can be achieved. The packing density ϕ is given by:
ϕ = π √3 / 6
Solution:
Fraction of Area Not Covered:
Complement of Packing Density: To find the area not covered by coins, we subtract the packing densityfrom1: = 1- ϕ = 1 - (π √3 / 6)
= 1- π/(2√3)
