__Category__: Solid Geometry, Plane Geometry"Published in Vacaville, California, USA"

Pass a plane through a cube so that the section formed will be a regular hexagon. If the edge of the cube is 2 units, find the area of this section.

__Solution__:

To illustrate the problem, it is better to draw the figure as follows

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The intersection of a cube with a cutting plane is a regular hexagon with 2 units of its sides. The vertices of a regular hexagon are located at the midpoint of six sides of a cube. By Pythagorean Theorem, we can calculate the sides of a regular hexagon as follows

Next, analyze the section as follows

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Since this is a regular hexagon, then we can calculate the vertex angle of the six triangles as follows

The six triangles are all isosceles triangles because the above figure is a regular hexagon. Let's calculate the base angles of an isosceles triangle as follows

Since all angles of an isosceles triangle are all equal, then all six triangles of a regular hexagon are equiangular or equilateral triangles.

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By using Pythagorean Theorem, the altitude of an equilateral triangle is

Therefore, the area of a regular hexagon which is the section of a cube is