Quandaries
and Queries
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.
Abraham,
The actual situation does not require you to know the radius.
So far, you have called the radius r.
You can use this same name again, and work out the RATIO of the length of the side and the length of the arc.
What you find is: length of edge = r, length of minor arc is pi(r)/3.
The ratio of these is: r / [pi (r) /3] This simplifies to 3/pi, as you hoped.
In a situation such as this, where you are seeking a ratio of two items which contain a variable, you could also explore the ration by plugging in one choice (or a couple of choices), to see what happens. For example, set r=1, and you get the correct ratio.
Double check that r=2 gives the same value!
Walter