Constructing A Golden Rectangle - Method One

Isn't it strange that the Golden Ratio came up in such unexpected places? Well let's see if we can find out why. The Greeks were the first to call phi the Golden Ratio. They associated the number with perfection. It seems to be part of human nature or instinct for us to find things that contain the Golden Ratio naturally attractive - such as the "perfect" rectangle. Realizing this, designers have tried to incorporate the Golden Ratio into their designs so as to make them more pleasing to the eye. Doors, notebook paper, textbooks, etc. all seem more attractive if their sides have a ratio close to phi. Now, let's see if we can construct our own "perfect" rectangle.

You will need a piece of paper, a pencil, and a protractor to complete this activity.

Note: to see how your figure should change, pass your mouse cursor over the images below. You should try to duplicate the figures outlined in black; the golden area is what your rectangle will eventually look like.

We'll start by making a square, any square (just remember that all sides have to have the same length, and all angles have to measure 90 degrees!):

Please note that everyone will have different size squares to begin with.

Now, let's divide the square in half (bisect it). Be sure to use your protractor to divide the base and to form another 90 degree angle:

Notice that we have made two rectangles.

Now, draw in one of the diagonals of one of the rectangles:

Measure the length of the diagonal and make a note of it.

Now extend the base of the square from the midpoint of the base by a distance equal to the length of the diagonal (the length of the diagonal should be equal to the distance from the midpoint of the OLD base to the edge of your NEW base):

Construct a new line perpendicular to the base at the end of our new line, and then connect to form a rectangle:

Measure the length and the width of your rectangle.

Now, find the ratio of the length to the width.

Are you surprised by the result? The rectangle you have made is called a Golden Rectangle because it is "perfectly" proportional.