Ruler and Compass Construction of a Square with Doubled Area

Mathematicians in ancient Greece showed that ruler and compass constructions could be used in a wide variety of geometrical operations. Some elementary examples are: drawing a line (or ray) through two points, drawing a circle (or arc) of arbitrary radius centered at a point, creating a point at the intersection of two nonparallel lines and drawing a perpendicular to a line at a given point. Three geometric constructions sought from antiquity are squaring a circle, doubling a cube and trisecting an arbitrary angle. These have since been shown to be impossible using only a ruler and compass.

Although doubling the cube cannot be accomplished by ruler and compass, the two-dimensional analog of doubling the square is possible [1]. This Demonstration outlines a nine-step procedure for constructing a square of area 2, starting with a unit square. Step 5 shows, in detail, the construction of a perpendicular line at a given point. The same procedure is implied in step 7 for constructing perpendiculars through P and D. The final result is shown in step 9: a blue square of side and area 2.