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Geometry - Dividing up a Surface

by Andy Slater

There are occasions when you need to divide a surface into equal widths, for example if you wanted to paint stripes on it. I've seen people struggle with this but having observed my father working with wood from an early age, the solution to this problem was practically second nature to me. Let's assume for example, he'd measured a piece of wood and found it to be 17 3/8 inches wide and he wanted to divide it into 4. Now he could have done the maths and tried to measure the result across the piece of wood but a much simpler method is to put the ruler diagonally across the wood so that he was measuring out a diagonal line 20 inches long. Marking the wood at 5 inch intervals is now simplicity itself.

Aris Kafantaris provided the following image to illustrate the process:

Geometry - Dividing up a Surface

Note that the diagram shows how the surface could be split into stripes by drawing lines that are parallel to the edges. An alternative would be to draw a second diagonal at the other end of the surface, split it into 4, and then join the 3 points on each of the diagonals together.

It's also worth noting that in the example shown, because we are splitting the width into 4, we could have marked out any diagonal line at random and used our technique for bisecting a line. However the beauty of our measuring technique is that we can use it to do some really 'difficult' divisions. In the example above we split a 20" diagonal into 4x 5" widths however we could have easily split it into 5x 4" splits. If we'd wanted to split the width into 3 we could have measured an 18" diagonal (and split it into 3x 6") and if we'd wanted 7 splits we could have used a 21" diagonal and measured out 7x 3". In a nutshell, we choose a length for our diagonal that will be easy to divide by the number of splits that we want.

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