A polygon of 100 sides is inscribed in a circle.

geometry/inscribed.polygon New

New entry, edited from material collected for the archive and not yet in a released version.

Problem

A polygon of 100 sides is inscribed in a circle.  If an angle is
inscribed into each of the 100 segments outside the polygon, the sum of
these angles in degrees is:

a)  Less than 2000
b)  Less than 1000
c)  Can't be calculated
d)  Greater than 2000
e)  Can be calculated only if the polygon is regular
f)  None of the above

Show solution

Imagine a regular polygon with movable vertices.

If the 100 vertices were to move so that the segments approaches length
0, the vertices converging together, there would be 99 inscribed angles
of 180 degrees, and the last inscribed angle would approach 0 degrees.

The value of the sum is:
99 * 180 + 0 = 17820.

The maximum distance between two vertices on a circle is 180 degrees.
If the vertices were divided in any way so that they converged over two
points:
In this case, the total sum of inscribed angles would be:
98 * 180 + 2 * 90 = 17820

If one place of concurrence is at 120 degrees and one at 240 degrees,
with the rest converging on zero (ie defining a regular "triangle", the
maximum is:
97 * 180 + 3 * 120 = 17820

If vertices converge at 90 180, 270 and the rest on 0, the maximum is:
96 * 180 + 4 * 135 = 17820

Moving a vertex from one point of convergence to another at any stage of
this yields a higher result, as the minimum sum of two inscribed angles
is at least 180.

This method allows the construction of all possible polygons of 100
vertices inscribed in a circle.

By series, it follows that the answer is
d)  Greater than 2000.

-- Paul Bishop <paulb@adinfo.com>

[ Editor: all four cases computed above give the same total, 17820, so
they do not show any case giving a higher result. ]