Number of Regions in Plane Defined by Given Number of Lines/Examples/6

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Example of Number of Regions in Plane Defined by Given Number of Lines

With $6$ lines, the plane can be divided into a maximum of $22$ regions:


LinesDissectingPlane-6.png


Proof

From Number of Regions in Plane Defined by Given Number of Lines, the maximum number $L_n$ of regions in the plane that can be defined by $n$ straight lines in the plane is:

$L_n = \dfrac {n \paren {n + 1} } 2 + 1$

Here $n = 6$, so:

\(\displaystyle L_6\) \(=\) \(\displaystyle \dfrac {6 \paren {6 + 1} } 2 + 1\)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {6 \times 7} 2 + 1\)
\(\displaystyle \) \(=\) \(\displaystyle 22\)

$\blacksquare$


Sources