Geometric Partition of a Circle Using Lines

Theorem: The maximum number of regions that the interior area of a circle can be partitioned into with n lines is f(n)=(n²+n+2)/2.

This essay provides a derivation of the formula for f(n) as well as a description of other geometric figures that this formula is valid for.

Start with a circle in a plane and let n be a nonnegative integer. The interior area of the circle will be partitioned into the maximum number of regions by n lines if and only if the following conditions are met.

• Each line must intersect the circle at exactly 2 points.
• Each line must intersect every other line in the interior of the circle.
• No more than 2 lines can intersect any single point.
• No more than 1 line can intersect any single point on the circumference of the circle.

Starting with a circle and no lines, add n lines one at a time in a way that partitions the interior of the circle into the maximum number of regions. Use the following rules to count the maximum number of regions that the interior of the circle can be partitioned into with n lines.

• The circle has 1 interior region when there are 0 lines.
• The 1^st line will increase the number of interior regions by 1.
• The 2^nd line will intersect the 1st line in the interior of the circle, increasing the number of interior regions by 2.
• The 3^rd line will intersect each of the previous lines in the interior of the circle, increasing the number of interior regions by 3.
• The n^th line will intersect each of the previous lines in the interior of the circle, increasing the number of interior regions by n.

Let f(n) be the maximum number of regions that the interior area of a circle can be partitioned into with n lines, where n is a nonnegative integer.

The formula for f(n) also computes the maximum number of regions that the interior area of some other geometric shapes can be partitioned into using n lines. The following is a basic description of the geometric shapes that formula f(n) will be valid for.

• Start with a flat plane and place the tip of a pencil on point A.
• Draw the outline of a geometric shape, returning to point A.
• Do not lift the pencil tip from the plane until the pencil tip returns to point A, at which time the figure is complete.
• Do not let the pencil tip touch any point more than once, except point A which the pencil tip will touch twice.

This geometric shape will have a single interior area that is not divided into more than one region. Shade the interior area red and perform the following test.

• If there exists a line that partitions the interior area shaded red into 3 or more regions, then formula f(n) is not valid for this geometric shape.
• If there does not exists a line that partitions the interior area shaded red into 3 or more regions, then formula f(n) is valid for this geometric shape.

The following is a partial list of geometric shapes that formula f(n) is valid for.

• Circle
• Triangle
• Square
• Rectangle
• Rhombus
• Parallelogram
• Simple trapezoid
• Regular simple pentagon
• Regular simple hexagon
• Regular simple octagon
• Regular simple polygon with m sides (for any integer m≥3)