Volume of a Box

This GeoGebra applet demonstrates the geometric calculation and mathematical modeling process of a box volume. Through the construction of points, segments, orthogonal lines, and polygons, the applet explores volume changes and optimization under specific conditions, using similar triangles and homothety transformations. The interactive model allows users to manipulate key geometric parameters and observe in real time how the volume of the constructed box varies. This applet originates from the Primary School Mathematics Creative Problem-Solving Competition and is designed to help students build intuitive understanding of solid geometry concepts. It integrates multiple mathematical themes, including piecewise functions and definite integrals, into a single cohesive exploration of volume calculation. Students can dynamically adjust dimensions such as the size of cutouts from a rectangular sheet and watch as the resulting box's volume changes accordingly. The applet guides learners through the full process of deriving volume formulas from first principles, constructing the relevant geometric figures step by step, and identifying conditions under which the volume reaches its maximum value. By combining visual construction with algebraic reasoning, the applet bridges the gap between abstract calculus-based optimization and tangible geometric intuition. Teachers can use this resource in classroom settings to illustrate how elementary geometry, algebraic modeling, and advanced techniques like integration work together to solve real-world optimization problems. The interactive nature of the tool encourages exploration and experimentation, enabling students to develop deeper conceptual understanding beyond rote memorization of formulas.