Question

pass a plane through a cube so that the section formed will be a regular hexagon. if the edge of the cube is 4cm . find the area of the section. help plss

Answer 1

If the plane intersection with the cube is a regular hexagon, then the plane intersects every side of the cube. Stand your cube on one of its corners, and visualize a plane cutting through it horizontally, midway between the top and bottom corners. If you look at the other six corners, the edges connecting each to the next form a zigzag shape going around the cube, and the plane cuts each of those edges at its midpoint.

So if the edge length of our cube is a (going for the general case), the plane's intersection with any face will be the hypotenuse of an isosceles right triangle with legs of length a/2, and therefore each side of the regular hexagon is the length of that hypotenuse: a(√2)/2.

The apothem (The apothem of a regular polygon is a line segment from the center to the midpoint of one of its sides. ) of the regular hexagon is, of course, (√3)/2 times the length of its sides--in this case,
a(√6)/4. The area of the regular hexagon is
6 * (apothem * side)/2
= 6*[a(√6)/4]*[a(√2)/2]/2
= 3a^2 (√12)/8
and for a=4cm, that's
6√12 square cm

Answer 2

thanks!