Geometric Dissection of a Cross into Two Squares

Geometric Dissection of a Cross into Two Squares This applet demonstrates a classic geometric dissection problem: transforming a cross-shaped polygon into two squares through area-preserving cuts and rearrangement. The construction uses GeoGebra tools including polygons, line segments, intersection points, and rotation to reveal how a single cross figure can be partitioned and reassembled into two separate squares without any loss or overlap of area. The applet allows students to interactively explore the dissection process step by step. Users can manipulate key points and sliders to control the cutting lines and observe how each piece moves into its final position. The dynamic visualization makes clear the underlying principle of equidecomposability: that two figures with equal area can always be divided into a finite number of congruent pieces that can be rearranged to form each other. Mathematically, the applet illustrates several important concepts in plane geometry. It demonstrates the conservation of area under rigid transformations such as translation and rotation, which are foundational to understanding geometric equivalence. It also provides a concrete example of how algebraic relationships between side lengths and areas manifest in a visual, intuitive way. Students can verify that the total area of the original cross equals the combined area of the two resulting squares. The construction highlights the elegance of geometric proofs that rely on dissection rather than algebraic computation alone. By watching the pieces slide, rotate, and lock into place, learners gain an embodied understanding of what it means for two shapes to be equidecomposable. This applet is well-suited for students studying Euclidean geometry, particularly those exploring topics such as geometric transformations, area formulas, and classical construction problems. It serves as both an instructional demonstration and an investigative tool, encouraging learners to ask questions about which cross shapes admit which square pairings and how the dissection lines are determined. It is appropriate for middle school through high school mathematics courses and complements lessons on proof, symmetry, and the relationship between geometry and algebra.