Question

A regular hexagon is inscribed in a circle of radius 23 inches. Find the length of the sides of the hexagon.
Answer in units of in

Answer 1

I gave you the key hint for this yesterday. Did you get it?

Answer 2

not at all.

Answer 3

Such a hexagon consists of six triangles, yes, each triangle having a vertex at the centre of the circle.

So each triangle must be isosceles, because two of the sides of each triangle are the length of the radius.

Answer 4

But now you realize that that the angle at the centre of the circle for each triangle must be 360/6 = 60 degrees. Therefore the two other angles of the triangle, which we know are equal, must also be 60 degrees.

Therefore each triangle is an equilateral triangle: each angle is equal, each side is of equal length.

Answer 5

Draw yourself a diagram and convince yourself of these facts.

Answer 6

Now that being case, you can figure out the length of one side of the hexagon, because we just showed that that side is exactly the same length as ... what?

Answer 7

the angle

Answer 8

No, way off.

Draw a circle and inscribe a hexagon in it, with the six triangles.

Answer 9

So here's a hexagon, with the six triangles.

Notice that the triangle OBC must be isosceles, because OB and OC are the same length.

Answer 10

But the angle at O must be 360/6 = 60, because there is 360 degrees in total in the circle and there are six equal triangles.

So now we know because the triangle is isosceles, that the angle at B = angle at C, write that as B = C. But as O + B + C = 180 degrees. But as O = 60 degrees,
B + C = 120 degrees. Yet as B = C, it must be that B = C = 60.

Therefore O = B = C = 60 degrees and the triangle OBC is not just isosceles, but equilateral.

Answer 11

Hence the side lengths, OB, OC and BC are also all the same length.

So what then is the length of BC?

Answer 12

so the radius of 23 inches has nothing to do with it?

Answer 13

The radius length has absolutely everything to do with it. The radius length is the length of OB and OC.