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Golden Rectangle Dimensions

November 12th, 2009 by Math Tricks | No Comments | Filed in Golden Rectangle, Mathematics Concepts

So you have a single line of length X, and you want to extent the line to height Y such that you produce a golden rectangle.  Simple as pi pie!

This problem can be solved with some simple algebra, and it is useful if you wish to draw a golden rectangle given a line of any length.  For instance, you may want to incorporate golden rectangles into some artwork you are working on, or you may wish to crop photographs such that they are framed within a golden rectangle.

So, given a line of any length, you can break the line into two parts:


It is easy to see that:


From this, you can calculate that the length (A) of the sides of the square part of the golden rectangle is:

A = (A + B)/1.618

So just extend the line into a rectangle with base=(A+B) and height=(A+B)/1.618

Using this type of reasoning, if you have a square with a side of length A, and wished to extend the length to A+B such that the A+B is the length of the base of a golden rectangle, you can determine the length of B very easily:

golden ratio

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Drawing a Golden Rectangle

October 21st, 2009 by Math Tricks | 2 Comments | Filed in Golden Rectangle, Mathematics Concepts

Drawing A Golden Rectangle

Drawing a golden rectangle is pretty easy using the technique I will show you here.  I used PowerPoint to draw the rectangle here, but you can use this technique to draw a golden rectangle with just about any drawing program.  If you have a compass and a protractor, then I suggest that you try to draw your rectangle on paper.

First, draw a square.  You do not need to know the dimensions of the square, just be sure that all the sides are of equal length:

making a golden rectangle 1

Next, find the midpoint at the base of the square, and draw a line from the midpoint to the upper-right corner of the square:

making a golden rectangle 2

Now draw a circle with the midpoint of the base as the center.  Expand the circle such that the line from the base midpoint to the upper-right corner is a radial line from the center of the circle to the circle’s edge:

making a golden rectangle 3Now rotate the radial line downward, keeping the edge of the line on the circle:

making a golden rectangle 4Continue with the rotation until the radial line is parallel to the base of the square:

making a golden rectangle 5Remove the circle.  What is left is the square with an extension at the base:

making a golden rectangle 6Draw a rectangle around the base + extension and square height:

making a golden rectangle 7Finally, remove the extension and square.  What is left is a perfect golden rectangle:

making a golden rectangle 8Very easy and neat.  Congratulations, now you can create great works of art by incorporating golden rectangles into you masterpieces!

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