Construct Golden Rectangle
This example demonstrates how to construct the golden rectangle using a compass and ruler. The golden rectangle is a rectangle whose sides form the golden ratio which is a naturally occuring number approximately equal to .
Steps

Start by drawing a square with a side length of 2. The length of 2 is chosen instead of a length of 1 because it will make the math easier later on.

Place a point in the middle of the bottom edge of the square so that the edge is divided into two equal parts.

Draw a line from the point to the top right corner of the square. This line forms the radius of the circle which we are about to draw. Using the pythagorean theorem we know that the length of the radius is . Also, extend the bottom edge of the square outwards in the right direction.

Using the point and the radius, draw the arc of the circle from the top right corner of the square in a clockwise motion until the arc intersects with the bottom line we extended.

Draw a perpendicular line up from the point of intersection and another line which extends the top edge of the square to get the top right corner of the golden rectangle.
We have now finished constructing the golden rectangle. The ratio of the sides form the golden ratio represented by the greek letter (phi).
We can also scale the golden rectangle by a factor of two to get a normalized version of the golden rectangle where the shorter side’s length is 1.