Sunday, May 4, 2014

More Cube Problems, 6

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

 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

 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.

 Photo by Math Principles in Everyday Life

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

The area of a triangle is

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