 # Ruler and Compass Construction of a Square with Doubled Area

Initializing live version Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

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.

[more]

Although doubling the cube cannot be accomplished by ruler and compass, the two-dimensional analog of doubling the square is possible . 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.

[less]

Contributed by: S. M. Blinder (August 2022)
Open content licensed under CC BY-NC-SA

## Snapshots   ## Details

Reference

 K. Brakke. "Ruler and Compass Construction Doubling the Area of a Given Square." (Sep 30, 2021) facstaff.susqu.edu/brakke/rulerandcompass/20-doubledsquare.html.

## Permanent Citation

S. M. Blinder

 Feedback (field required) Email (field required) Name Occupation Organization Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback. Send