Start by taking angle x to be measured in radians. Then the length of arc AC is exactly x. What you want to show is that . Notice that you have two right triangles- one, triangle OAB, with hypotenuse of length 1, the other, triangle OAD, with a leg of length 1.
Yes, I know all of that - I drew the diagram. I'm trying to find an (hopefully intuitive) geometrical argument that the length of the arc is between the lengths of the two line segments.
because is the shortest distance between the line and the point . But I can't think of a good reason why the arc must be shorter than . There are certainly curves within the triangle that are longer than , so how can I argue (without reference to and ) that the arc is shorter than the line segment ?
Extend the green arc around to make a circle. There are five more equilateral triangles of the same size that are adjacent to each other, all having point O as a
vertex, and beginning with a triangle that shares a side with line segment OD directly below.
The six equilateral triangles overlayed on the green circle divide the circle into six congruent sectors. The altitudes of each of these equilateral triangles divide
each sector in to two smaller congruent sector pieces. The green arc in the problem is associated with one of these smaller sector pieces. There are 12 of them.
The length of each green arc for each smaller sector piece, radius equals 1, is (2*pi*1)/12 = pi/6 ~ and so that is what arc AC's length is approximately.
The outside figure is a regular hexagon. Look at one of its sides. But line segment AD equals half the length of one of its sides. Triangle OAD is a right triangle,
and it's an equal half of an equilateral triangle. Then it's a 30-60-90 triangle. Because the altitude equals 1, then the shortest side equals 1/sqrt(3). That shortest
side is line segment AD.
The shortest side of triangle OAD is line segment AD, and its length equals
Therefore, line segment AD is longer than arc AC.