More Cube Problem, 7

Category: Solid Geometry, Trigonometry

"Published in Vacaville, California, USA"

The plane section ABCD shown in the figure is cut from a cube of edge a. Find the angle which the section ABCD makes with the lower base of the cube if D and C are each at the midpoint of an edge.

Photo by Math Principles in Everyday Life

Solution:

To understand more the problem, it is better to label further the given figure as follows

Photo by Math Principles in Everyday Life

Since the edges of a cutting plane which are AB and CD are parallel to the four parallel sides of a cube, then two triangles formed by a cutting plane with a cube are congruent. 

Since the edges of a cube are perpendicular to each other, then two triangles are right triangles.

Therefore, the angle of a cutting plane with respect to the lower base of a cube is

 

or

 

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

More Cube Problems, 5

Category: Solid Geometry

"Published in Vacaville, California, USA"

One cube has a face equivalent to the total area of another cube. Find the ratio of their volumes.

Solution:

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

Photo by Math Principles in Everyday Life

Let B = be the area of a base of a large cube
      x = be the length of the edge of a large cube
      T = be the total area of a small cube
      y = be the length of the edge of a small cube
      V₁ = be the volume of a large cube
      V₂ = be the volume of a small cube 

The area of a base of a large cube is

The total area of a small cube is

 
From the given problem statement, we know that
 

 

The volume of a large cube is

 

 

 

The volume of a small cube is

Therefore, the ratio of their volumes is