Help Mathematicians! Please Solve this problem! What is the area of the regular hexagon?

Question

The intersection of a unit cube and a plane passing through its center is a regular hexagon. What is the area of the regular hexagon? Thanks for your help.

Answer 1

hi!
you just have to split the regular hexagon into 6 equilateral triangles.
if you want the exact formula, please visit
http://www.drking.plus.com/hexagons/misc/area.html
hope to help you though i know for myself that i am not a mathematician and just a high school student.
~_~

Answer 2

That's tricky, because the real issue is figuring out what angle the plane cuts, what the sidelength of the hexagon is and so on.

If we know the sidelength, finding the area is easy -- just divide it into 6 equilateral triangles, bisect each of the triangles again if you like, and either look up a formula or (my preference) do a very small amount of trig.

As for figuring out exactly how the plane cuts the cube -- you can start by drawing pictures. But before long you'll be algebraically calculating where the plain hits the edges and, hence, how long the line segments of intersection with each face are. When they're equal, then

A. You have the hexagon
B. You just calculated its sidelength.