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Math is fundamental to building. And while there is no escaping the need to be proficient with numbers and algebra, learning some geometry can go a long way in helping carpenters be more efficient. In this article, we’ll introduce a few methods for dividing, measuring, and laying out shapes that can save time and take some of the head scratching out of many layout jobs.
Scale. The first principle we’ll address is the idea of using a scale of known increments (such as a tape measure) to divide a line or area. In the first two examples below, we don’t need to know the width of the board we want to divide, or make any calculations. We simply hold the end of tape along one edge and run it out to the opposite edge. The distance and angle of the tape don’t matter, as long as we align the opposite edge with a number on the tape that we can easily divide by 2 or 3 (or any divisor we want). This same concept can be used to divide an uneven distance into a number of equal segments without having to pull out a calculator.
Angles are fundamental to joinery. Most of the time, we are dealing with right angles (90 degrees) and regular miters (45 degrees). What happens when we encounter other angles? Our inclination is to measure that angle in degrees somehow (with an electronic angle finder or a protractor, for example) because the table on a miter saw is set up with degree adjustments. However, a lot of times we don’t need to know the numbers. The miter for joining boards of unequal width, for example, is an acute angle less than 45 degrees, which we can lay out using parallel lines, as shown below. Once we understand this idea, we can just measure the width of one meeting board off the end of the other and join the diagonal.
Or we can lay out any angle using a compass and a straightedge to bisect the angle as shown below.If we do this on a scrap piece of wood, we can cut the angle we’ve drawn by aligning it with the blade on our miter saw—no numbers needed.
The same concept of swinging an arc (either with a compass or with a tape measure) can also be used to lay out different shapes, such as a square, a hexagon, or an octagon, as shown in the illustrations below.
Leveraging the Pythagorean theorem. When we frame roofs or square large areas like a floor frame or lay out plates in preparation to stand walls, we can’t avoid the Pythagorean theorem: a2 + b2 = c2. If the abstract nature of this formula makes you nervous, it may be helpful to see the equation spelled out in geometric form, as shown below. (This visual explanation also gives us a construction of the square root function that maps the area of a square to its side length.)
The example shown above is a 6-8-10 right triangle—a larger form of the all-powerful 3-4-5 right triangle; if one leg of a triangle measures 3 and the other 4, we know by the Pythagorean theorem that the diagonal measurement will be 5. But we don’t have to calculate this. Rather, we can use these dimensions as a check for square for any right angle: By measuring out one side equal to 3 units (inches, feet, anything) and the other side equal to 4 units, we know the hypotenuse connecting these sides will equal 5 units—or any multiple of 3-4-5: 6-8-10, 9-12-15, 12-16-20, and so on. When you’re squaring a floor or a foundation (see example, below), it’s always most accurate to use the largest multiple for the walls that you are laying out. For a more detailed explanation of this layout procedure, see "Framing Square Basics: Foundation Layout."
The Pythagorean theorem relates to more than roof framing and right-angle layout; it can also be used to define the ellipse created where a vent pipe passes through a sloped roof. When the roof deck is part of the air barrier, we want to create a hole for the vent that can be tightly sealed.