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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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| Photo by Math Principles in Everyday Life |
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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| Photo by Math Principles in Everyday Life |
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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| Photo by Math Principles in Everyday Life |
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
