Some miniatures with a cube

A detailed solution to several problems with a cube is given including a number of examples related to the calculation of maximum volumes for symmetrical shapes inscribed in a cube such as an equilateral cylinder a regular cone and a sphere. For example, in one of the problems with a sphere it was necessary to find the volume of the outer part of the sphere that goes beyond the side faces of the cube. All the examples considered have one goal: to teach high school students to navigate stereometric problems as one of the most difficult parts of elementary mathematics contributing to the development of their spatial thinking. All the examples given in this paper are aimed at this and the results of all calculations use terminology well known to schoolchildren as well as the cosine theorem which is necessary for solving a number of specific examples. At the end of each solved problem brief answers are given the wording of which is concise indicating the completeness of the corresponding section. The solutions of all problems are accompanied by detailed drawings illustrating their formulation and all the spatial figures depicted clearly explain the essence of each of the tasks. The paper would be useful for mathematics teachers in secondary education and pedagogical institutions teaching a stereometry course for high school students. All the given examples should help with the material due to the detailed solutions provided for each of the problems given. First of all, such necessity for its coherent treatment is focused on developing the sparial thinking of the high school students. This would be very useful for their studying of more complicated branches of mathematics in the future.