Question

what is the maximum number of sections into which a circle may be divided into by drawing four straight lines?

Answer 1

11 sections

Explanation:

This problem is called the circle cutting or pancake cutting problem.

Let the number of cuts or divisions by straight line = n

With this information it is possible to calculate any number of pieces or section a circle will be divided into what straight lines are drawn (cut) across the circle.

When a straight line is drawn across the circle, it divides the circle into 2 sections or regions. The nth straight lines will divide the circle into n new sections or regions, so the progression is;

f(1) = 2

f(2) = 2 + f(1)

f(3) = 3 + f(2)

.

.

.

f(n) = n + f(n-1)

Therefore,

f(n) = n + [(n-1) + f(n-2)}

= n + n-1 + ... + 2 + f(1)

= f(1) + ∑
`_(i = 2)^(n)`i

=
`2 + (1)/(2) (n + 2) (n - 1)`

=
`(1)/(2)(n^(2) + n + 2)`

When n = 4

=
`(1)/(2)(4^(2) + 4 + 2)`

= 22/2

= 11 sections