irregular shaped hexagons, biggest inside circle and center point?

Question

How can I construct the center and the biggest possible inside circle of irregular shaped convex hexagons. I would like to do this using a geometric based workflow.

Answer 1

Geometrically speaking, find a point that has equal distance from the two closest points. Once found, make this point be of equal distance from second closest points. Continue until this point is inbetween all the points in such a way, that if moved slightly, it will destroy the balance it has. This point will be the center of the circle. Now use your compass, and draw a circle with this point, that stretches out to the closest point.

Answer 2

I don't know what a geometric based workflow is, but this is what I would do.

Draw the bisectors of each of the hexagon,s interior angles.
Starting with the largest angle, select a point on its bisector and draw a circle that is tangent to the sides of the angle. If part of the circle is outside the hexagon, move the point closer to the vertex and try again. Continue this process, until you have the largest circle that fits within the hexagon. Measur the length of the radius.

Repeat this process with the other five bisectors. Pick the circle with the largest radius.

You might also try the same thing with the perpendicular bisectors of each side, But I have a gut feeling the angle bisector method has the best chance of success.

Repeat this procedure

Answer 3

There is not a guaranteed unique circle, although there is going to be a maximum area. Your question asks for the circle location, but since uniqueness is not assured, does that answer your question inadvertently?

To see that uniqueness wont exist, visualize a regular hexagon, fixed three consecutive vertices in place and the stretch the other 3 'straight away' making a long tubelike hexagon. Lots of different circles can fill that with the same maximum area.

Answer 4

Find the diameter of the hexagon. the D of the hexagon divided by 2 will give you radius.
then use the radius to get the largest area of circle within hexagon
r^2pi

then get the area of the hexagon
you can find the formula here
http://en.wikipedia.org/wiki/Hexagon

Answer 5

Because it's irregular it makes things difficult.
Try this:
Join all the opposite corners together. Measure these straight lines. Take the longest one, and bisect it. Use the bisected point as your centre point.
Now by trail and error, adjust the compass and draw the circle to your required size.