Pyramids and Cubes: Developing Spatial Imagination

This material is suitable for older school-aged children.

We begin studying geometry with planimetry, thoroughly examining geometric shapes on a plane. However, the world around us is three-dimensional, and it’s better to start introducing polyhedra and solid figures from an early age. This section of geometry is called stereometry. Solving stereometric problems often involves reducing them to a flat model. Many objects in stereometry have counterparts in planimetry, such as a sphere and a circle, or a cube and a square. To understand the nuances of stereometry, it is important to have a good grasp of spatial figures and be familiar with the basic concepts of planimetry.

During lessons, it’s essential to use models of polyhedra and round bodies. Visually impaired students explore objects through touch. The exploration should be done gradually, piece by piece, and slowly. The student should understand the shape of the object and the structure of its surface. An effective tactile exploration process involves using both hands, where one hand is often the searching hand and the other is controlling. Touch provides the most accurate representation of the object, and words complement the tactile image. The teacher must continually explain the material during the lesson. However, it is not enough to provide verbal descriptions of objects for visually impaired students; they must work with models of polyhedra.

A polyhedron is a solid body bounded by several surfaces (polygons). For example, a pyramid or a cube.

Here’s a question: How many identical quadrilateral pyramids can a cube be divided into? In class, we assemble quadrilateral pyramids, then lay them out in the form of a cube’s net. The result is a cross made of six pyramids. When the net is assembled, all the pyramids meet at their vertices in the center of the cube.

Conclusion: A cube can be composed of six identical quadrilateral pyramids, whose common vertex is at the center of the cube, and whose bases are the cube’s faces. The cube is directly related to the number 6—six faces.

Problem: The volume of the cube is 72. Find the volume of a quadrilateral pyramid whose base is a face of the cube and whose vertex is at the center of the cube.

Solution: There’s no need to derive a formula; simply use the fact that six pyramids fit into the cube. Answer: 12.

Creativity space: Let’s attach six equal pyramids to the cube, as we did before when forming the cube. This creates an interesting transformation. Each pair of triangles from different pyramids lies on the same plane, forming a single rhombus-shaped face. This polyhedron is called a rhombic dodecahedron, which has 12 rhombic faces.

Question: How many and what kind of polyhedra should be “cut” from a cube to form a tetrahedron?

A tetrahedron is a triangular pyramid. To answer this question, it’s best to use a plasticine cube to make various cuts. We make a cutting plane through three vertices of the cube. As a result, we get a triangle and a triangular pyramid is cut off. Repeat this process three more times, and we end up with four triangular pyramids. The pyramids are identical. In the center of the cube, after the cuts, a triangular pyramid remains. The pyramid has four vertices, all edges are equal, and its faces are equilateral triangles. This pyramid is a tetrahedron.

During such lessons, students acquire specific mathematical knowledge necessary for practical application.