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 occurring number approximately equal to .

Start by drawing a square with a side length of 2. A 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’s 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 that extends the top edge of the square to get to the top right corner of the golden rectangle.

We have now finished constructing the golden rectangle. The ratio of the sides forms 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.