Apr 29 2008

The Constructs of Squareness

Published by at 11:30 pm under Layout Strategy,Metrology

The prevalence of the right angle in engineered structure is probably second only to the straight line in order of importance. Engineered structures in wood are often using plane geometry to help describe and document what they are and how to build them.

Much of the way we think about civil engineering, architecture, woodworking, and even some metalworking, call it flat work if you like, is based on previously understood, maybe even taken for granted, notions about geometry.

Every line that goes in a given direction without variance to that direction is straight, all points that lie upon a line, line segment, or ray can be thought of as congruent. At any point on a line, another line, line segment or ray can intersect, begin sharing a common end point, and create an angle.


There are four ways we look at angles… The most basic angle is the right angle, the angle of 90 degrees, which when measured, corresponds to a quarter of the 360 degrees in a circle, or some thing other than a circle that circuitously begins and ends at the same point. The other ways we describe angles are of angles smaller than 90 degrees which are “acute” and angles larger than 90 yet smaller than 180 degrees, which we call obtuse. If the angle is greater than 180 and less than 360 degrees we call it a reflex angle. When working in terms of squareness, we are only concerned with the 90 degree, or right angle.

Classically, a square has four angles and each of those angles is 90 degrees. If we add all four of those angles together, the result is 360. The interesting thing to note here is that in geometry, and fine work, 360 is not acceptably 359 or 361, and considered a fit. It is either square, or not.

Unlike all the other geometric shapes that use right angles, the square has four sides that are of equal length. This gives us two diagonals, which are also equal. When the diagonals are equal, they are equal to 1.41 times the length of the sides, otherwise known as the square root of two, and this value is referred to as Pythagoras’ constant. These diagonals also form the hypotenuse of right triangles, if the sides of said triangles are equal length.

Now, making your head hurt is not what I am trying to do, but you now know that you can check for squareness if the diagonals within the square are equal length. But what if the sides are not equal length? Well if 2, 3, or 4 sides are not equal, then you don’t have a square, and the angles will not be 90 degrees, except in one case, and that is when each pair of opposite sides of the 4 are equal length, yet adjacent sides are not equal length, This too creates square corners, can be checked with equal length diagonals, and Pythagoras’ Theorem is used instead to find the length of the diagonal. and conversely

Did you guess? This squared, non square is called a rectangle.

Square to the builder is simple, it can be the box or the rectangle, but it is most usable as another name for a right angle. Cutting something to square, or squaring something simply means to form an accurate right angle on the end of it.

With that and a tape measure you can square boards, boxes… Power at your fingertips!

The creation of square where there is not square is easy to do, positioned wherever you need it. You need sharp pencil, a ruler and a compass. Follow along with the diagram, hand drawn by the way, just to show that it can be drawn anywhere. Here are the steps:

With the ruler, draw line AB, and make point A on the left end. With the compass point positioned on point A, swing a short arc mark at any radius length you like, roughly off to the right of the intended perpendicular near the 45 degree radian. Pick any spot you like on that small arc line and cross it to mark your RP, or Radius Point.

With the same compass setting, transfer the point of the compass to the RP and starting at point A, draw a circle based on this unaltered radius length.

At the point where the circumference of the circle intersects line AB, establish point B.

Place the pencil at the RP and position the ruler against it. Align the ruler through the RP and point B. Then draw a line through the circle that intersects point B, the RP, and the circumference opposite point B. Establish this new intersection as Point C. This is Line BC.

Position your pencil on point C, and position the ruler against it. Align the ruler to point A, and draw line AC.

Depending on the how and where this is arranged, this creates a right angle every time. Layout lines for square. The process is called Erecting a Perpendicular. Simple, accurate, scalable, uses few tools. Squareness wherever it is needed. You can even draw it upside down and backwards.

Perhaps line AB and AC already forms the edge of a board or panel. That works. If point A is allowed to be the corner, a point B is established along the horizontal, and a Point C is established along the vertical side, and measurements are taken of line AB and AC. Plugging these measurements into Pythagorean theorem, and conversely will give the length of the hypotenuse, and if it does not measure the same, the board or panel is not square.

In any case, the constructs of square have some pretty simple and humble beginnings, and when observed in working, they help things fit. Now we know a bit more of the back-story. The fit and finish of any project is often defining of many things. Squareness often plays it’s part in the mix, and is often what we are striving for.

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The list of tools we make is too long to list here! Come to the Evenfall Studios Woodworks Store to see our latest New Tools, or have a look at our entire woodworking tool line!

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© Copyright 2008 by Rob Hanson for evenfallstudios.com All Rights Reserved.

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