Expert: Josh - 12/8/2004

Question
A regular hexagon is inscribed in a circle. What is the ratio of the length of a side of the hexagon to the minor arc that it intercepts?
(1) pi/6
(2) 3/6
(3) 3/pi    (This is the correct answer.)
(4) 6/pi
I found the length of the minor arc to be (pi)(r)/3 by doing a sixth of the circumference(2pi r).But I can't find the length of the radius to finish off the problem. If I knew the radius I would then plug it into the above and then use the radius again to be the length of the side because the triangle(one of the six of a hexagon) is equilateral. But can you show me how to get the radius to be 3? Thank you so much.


Answer
Hi Jeff,

Your reasoning for the length of the minor arc is correct. I think it helps if we derive things in general terms, and not assume a particular value for the radius.

Let the radius be "r".
I am going to ask you to draw a diagram now, to prove this geometrically.

Step 1: Inscribe a hexagon inside a circle (use a protractor, compass or whatever).
Step 2: Label each of the vertex (corner) of the hexagon as point A,B,C,D,E,F in clockwise order.
Make sure that all six sides AB, BC, CD, DE, EF, FA are connected.
Step 3: Label the midpoint between point A and point B, point G.
Step 4: Label the center of the circle O.

Now, consider the segment or triangle described by OAB.
In particular, the side |AB|=|AG|+|GB|.
We already know that |OA|=|OB|=r, the radius.

Fact 1: A perpendicular bisector (i.e., OG) divides the angle ^AOB into half.

Fact 2: It is known that ^AOB=(2*pi)/6=pi/3.

Therefore, ^AOG=(^AOB)/2=pi/6 or 30 degrees.

Using trigonometry, length |AG|/|OA|=sin(^AOG), right?
[taking the opposite side over the hypotenuse in triangle AOG]
So, |AG|=|OA|*sin(^AOG)=r*sin(pi/6)=r/2.
Finally, |AB|=2*|AG|=r.

Therefore, the length of side AB divided by the length of arc AB is given by r/[(2*pi*r)/6], which equals 3/pi.

It works for any radius.

Let me know if anything bothers you.

Cheers.