Construction of Polygons

In geometry a 'construction' has a strict meaning: you may use only a ruler (straight edge) to draw lines and a compass to copy lengths and draw arcs. You are not allowed to measure angles with the protractor scale or measure lengths with the ruler's markings inside the figure itself, because the aim is to build the shape from pure geometric rules rather than by eye. Every length is carried from one place to another by opening the compass to that distance, and every special angle is built from arcs. The reward is a figure that is exact, not approximate.

To construct a polygon you need just enough information to fix it and no less. A triangle is determined by three measurements - three sides (SSS), two sides and the included angle (SAS), two angles and the included side (ASA), or the right angle, hypotenuse and one side (RHS). A quadrilateral has more freedom, so it needs five independent measurements: for example four sides and one diagonal, or three sides and the two angles between them. Knowing this stops you from getting stuck halfway, wondering which point to fix next.

Many constructions need a fixed angle. A 60 degree angle comes free: draw an arc from the vertex cutting one arm at P, then with the same radius cut that arc from P; the line to the crossing point makes exactly 60 degrees, because you have effectively built an equilateral triangle. Bisecting that angle gives 30 degrees. A perpendicular (90 degrees) is built from a perpendicular bisector, and bisecting it gives 45. So 60, 30, 90, 45 and 120 degrees are all reachable with the compass alone, while other angles need a protractor.

Special quadrilaterals carry hidden information. In a parallelogram opposite sides are equal and the diagonals bisect each other; in a rectangle every angle is 90 degrees; in a rhombus all four sides are equal; in a square both. These properties hand you extra measurements for free, so you often need fewer given facts. Behind every step is a locus: the corner you want lies a known distance from two earlier points, so you draw an arc (a locus of points one distance from a centre) from each, and where they cross is the only point satisfying both - the corner.