Getting Things Square With the World: 3-4-5 Triangles

Construction projects often need to have precise 90 degree or “square” angles. But often the available tools (such as a carpenter’s square) are simply too small to guarantee the accuracy needed for large projects such as laying out the foundation of a house. So carpenters and concrete formers will often employ a 3-4-5 triangle technique to ensure accurate 90 degree angles.

The technique simply requires that a person create a triangle in the corner of the lines that are to be square (90 degrees) to each other. The triangle must have one side (leg) that is 3 feet long, a second side that is 4 feet long and a third side that is 5 feet long. Any triangle with sides of 3, 4 and 5 feet will have a 90 degree angle opposite the 5 foot side. If a larger triangle is needed to increase accuracy of very large structures, any multiple of 3-4-5 could be used (such as a 6-8-10 foot triangle or a 9-12-15 foot triangle).

So, mathematically why does this technique create a perfect right angle??

In Geometry, a well known method of constructing a right angle is to employ the Pythagorean Theorem. The mathematician, Pythagoras, discovered a relationship between the sides of any right triangle that is now known as the Pythagorean Theorem; he proved that the square of the longest side (the hypotenuse) is equal to the sum of the squares of the remaining two sides. This is often expressed as the following equation:

where A and B are the two legs of the right triangle and C is the hypotenuse. If we substitute the numbers from a 3-4-5 triangle into this formula, we then have:

Of course any lengths could be used to create the right angle for construction – as long as they were correct when applied to the Pythagorean theorem. But practically speaking, most other numbers would not work well. First of all, finding a square root on the job site would often require a person to carry a calculator around. Secondly, once a square root was found, it often could not be accurately located on a tape-measure or other measuring tool.

For example; suppose the carpenter chose to use triangle legs of 6 and 7 feet. Using the Pythagorean theorem we would find:

Solving for C would produce a hypotenuse of 9.2195444 feet – which would be very difficult to calculate mentally or locate on a tape-measure that is graduated in 8ths or 16ths of an inch.

Therefore using triangle dimensions of 3, 4 and 5 is easy to remember (no calculations needed), will always produce a perfect right angle and is easily found with common measuring tools.

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A Parallel Universe…

Centering Photographs on a Matt

A common task in photography or graphics is to center and attach a photograph or document to a matt or tag-board background. The technique often used to accomplish this is to place the photograph on the matt and align it with one of the corners (i.e. as in the image below, placed in the lower left-hand corner).

In this example, measurements are made and marks are placed half-way between the picture and the opposite edges of the matt (in this case, the right and top edges of the matt). The problem often encountered in this process is that the distance from the edge of the picture to the opposite edge of the matt is seldom a distance that is conveniently divided in two when finding the half-way points.

A slick trick…
Photographers and graphic artists have developed a simple procedure to overcome this problem by placing the ruler at some angle other than 90 degrees to the edges of the picture and matt. The angle of the ruler is adjusted so that the ruler crosses the edge of the picture on a whole-inch mark and at the same time crosses the edge of the matt on a different whole-inch mark – preferably with an even number of inches between those intersections. The mid-point or halfway point is now more easily calculated and marked. This is done twice along one side of the picture (right side in our example) and once along the remaining side (top in this case). Three small marks are placed to establish the center locations and the picture is aligned with them. In practice, these marks are made to be barely noticeable so that they do not show when the picture edge is aligned with them. Trying this technique a few times, you will find it a very quick and accurate solution to centering a picture.

The mathematics behind this technique: Applied geometry…
Mathematically, the task is to center a smaller rectangle inside a larger rectangle.

Establishing two marks on the right side of the picture:
In this picture centering exercise, once the picture is aligned with a corner of the matt, the opposite sides of the picture and the opposite sides of the matt form parallel lines (technically, parallel line segments: lines by definition extend infinitely in both directions, and since the edge of the picture has a fixed length with two end points, by definition it becomes a “line segment”). The edge of the ruler represents a “transversal” (a third line) intersecting the two parallel line segments. Calculating the midpoint of the transversal is referred to mathematically as finding the transversal bi-sector. Furthermore, one of the well-known geometric postulates is the fact that the bi-sector of a transversal is also the mid point between the two parallel lines.

This task requires establishing two transversals (AB and CD) and two bisectors (M1 and M2) between the parallel line segments of the right sides of the picture (AC) and matt (BD). In geometry, two points are all that are required to establish a line, and moving the edge of the picture to align with the two bi-sector points (points M1 and M2 in the figure above) essentially makes the edge of the picture a line segment passing through the two points. Additionally, since the two points M1 and M2 are equidistant from the original parallel line segments, the edge of the picture is now parallel to the outside edge of the matt.

Finally, sliding the picture up to the third mark: Why is only one mark needed on the top?
The top of the matt and the top of the picture are now also parallel to each other. This can be proven by the fact that the top of the matt and the top of the picture are “perpendicular” or at right angles (90 degrees – by definition of a rectangle) to the two original parallel line segments. And lines that are perpendicular to parallel lines will in fact be parallel to each other. Since the top of the picture and the top of the matt are already parallel to each other (equidistant from each other), it is only necessary to slide the picture up to a single mark (M3) that will leave an equal distance between the top and bottom edges of the picture and matt.

One last note of practicality:
It is always wise to minimize the impact of errors that might occur when measuring. By selecting whole inch marks on the ruler to serve as the intersection points for the transversals, this technique reduces the frequency of division mistakes by allowing the photographer/graphic artist to select numbers (preferably even whole numbers of inches) that are easily mentally divided in half. Even so, the process of transferring the measured location to a mark on the matt includes the potential for a small amount of error. Therefore the marks on the matt (M1 and M2) should be made as far apart as possible. That will minimize the degree to which the edges of the picture and matt will be out-of-parallel if a mark is slightly off of its correct position.

The Slippery Slope of Pitch…

Slope is a mathematical concept that is critical to much work in Algebra and other courses. The equivalent concept in Building Construction is Pitch.

In the construction world the concept of pitch is most often applied to the angle or slope of a roof and is defined as “rise over run”. In other words the pitch of a roof is actually a ratio, expressed as a fraction, with the numerator being the rise and the denominator being the run.

So a roof that “rises” 5 inches vertically in a “run” of 12 inches horizontally, is referred to as a 5/12 pitch roof.

This seems fairly straight forward and one might assume that a storage shed roof that rises 2 feet over a run of 4 feet would be described as having a 2/4 pitch. However, many occupations employ “conventions” or standard ways of proceeding or communicating. One of the conventions regarding pitch in the construction trades is that the pitch is always converted to have a run of 12 inches. In mathematical terms this means that any pitch with a denominator other than 12 must be converted to an equivalent fraction (ratio) with a denominator of 12. Therefore in the example above, the storage shed roof with a pitch of 2/4 would be converted to a fraction or ratio of 6/12.

The Pitch of a roof may be measured by using a level and a tape measure, either from the surface of the roof or the rafters within the attic. Both methods use a level to establish the run line and the tape measure to measure the rise. Both of these procedures are illustrated in the pictures below.

Measuring pitch from roof surface: measure down to the shingles from a 12″ mark on the level.

Measuring pitch when in attic: Measure up to the rafter from a 12″ mark on the level. More information on actual measurment techniques may be found at: http://www.improvenet.com/HomeOwner/ProjectTools/Calc/roof.html

An additional convention in the construction trades is to describe the pitch by stating the numerator (rise) first and the denominator (run) second. In other words the pitch of the storage shed roof would be stated as “six-twelve” or “six in twelve” – not in standard fractional terminology, which would have been “six-twelfths”.

How then, are pitch and slope alike?

• Both terms describe the steepness of a sloped line or surface.
• Both terms are basically describing the steepness of the hypotenuse of a right triangle created with the x-axis representing run and the y-axis representing the rise.
• Construction conventions always place the rise over the run as a fraction describing pitch; mathematics conventions always place y over x as a fraction in describing slope.

How are pitch and slope different?

• Pitch is always described as a positive number; slopes can be negative or positive.
• Pitch is always converted to have a denominator or run of 12; the x factor in the slope ratio can be any real number (whole numbers, decimals, negative or positive).

Hello world!

I’ve created this blog as a resource for teachers and students of mathematics and technology education. The primary goal is to regularly pick a technology topic and explain the underlying mathematics. I welcome the comments and contributions of anyone with experience in the topics presented, and together I hope that we can provide a context for teaching mathematics and a solid connection to the mathematics behind technology applications.

Thanks for visiting and I hope you return often!

Mike