Constructing the Cube Root of Two

This Demonstration shows a construction of using a ruler with two marks (red and black ) at a distance 1 unit apart, constrained to lie on the axis and the blue line. Drag point until the straight line touches the hexagon at . Then .

Explanation: The slope of is and the slope of the blue line is . Since their product is , the blue line is perpendicular to the line . Therefore is a right-angled triangle, so . The angle is , so . Since triangles and are similar, . Therefore , substitute for to get , and clear the denominator: . Expand to get , rearrange terms to get , and factor , ; and take the positive solution .