Mathematics of carpentry

Summary: Precise measurement is the foundation of the building trades.

While the word “carpentry” originally comes from the Latin root for chariot maker, today, the term refers to a number of trades that use wood for the construction of buildings and other articles. As there is a wide range of activities involved in carpentry tasks, carpenters must possess many different manual and intellectual skills to function in the profession.

Types of Carpenters

Carpenters who work on houses often fall into one of two broad categories: framing carpenters who work on the rough frame of a building, and finish carpenters who complete trim, stairs, railings, shelves, and other detail work. However, in practice, many carpenters end up doing some of each type of work, and carpenters who specialize in remodeling may not only do framing and finish carpentry but also tasks that are not strictly carpentry at all, such as plumbing, wiring, sheetrock finishing, and painting. There are also carpenters who specialize more narrowly, such as cabinet makers or carpenters who work on the specialized joinery between large posts and beams required in timber frame and log cabin construction.

Tasks of the Carpenter

Carpentry requires a variety of skills, including reading blueprints, measuring, cutting, fastening, and finishing. In addition, a carpenter must have knowledge of materials, including a variety of wood products and fasteners; and tools, including measuring devices, saws, drills, hammers, planes, and sanders. Carpenters who work on their own or as subcontractors on larger jobs must also have skills in cost-estimation and billing.

Consider, for instance, a carpenter who has been hired to add a covered deck onto a house. This carpenter might begin by working with the homeowner to determine the size and shape of the deck, possibly using a Computer Assisted Design (CAD) program to generate three-dimensional representations of how the finished project will look. After deciding on a design, the carpenter will need to use structural engineering tables to assess structural issues related to the design, such as the dimensions required for posts, the placement and size of cross-bracing, and the sizes of timbers that will be needed to span the distance between posts. From the calculations, the carpenter will then generate a price estimate, based on a materials list and an estimate of labor. The actual construction will include pouring concrete footers for the posts, measuring and cutting posts and joists with a circular saw, fastening materials to one another and to the house, screwing decking materials to the framing, framing a roof, installing roofing materials, constructing a railing, and building and finishing a set of stairs from the yard to the deck.

A Carpenter’s Calculations

In the process of creating a simple covered deck, this carpenter will be making many measurements, calculations, and decisions regarding:

Layout: The initial position of the deck must be laid out so it is square to the house. To do this, the carpenter will construct a set of batter boards that are set outside the corners of the proposed deck and allow strings to be pulled to mark the edges of the deck. Employing the rule that the diagonals in a rectangle are equal to one another, the carpenter adjusts the strings to bring the corners to 90 degrees. Corner square may also be established and checked using the Pythagorean theorem.

Footers: Each post will be anchored to a concrete footer that will prevent it from moving or sinking into the ground. The bottom of the holes for these footers must be dug below the freeze level for the geographic area where the deck is being built so that the footers will not be heaved out of place by the freezing and resulting expansion of the soil. By consulting the building code, the carpenter will determine the appropriate area for the footer in square feet; multiplying by the height will give the cubic feet. If this is a large project, where the concrete will be delivered, the carpenter will have to convert cubic feet to cubic yards, as this is the unit in which concrete is ordered.

Raising the Posts: After pouring the footers, the carpenter will raise the posts for the deck being built. Since these posts will also support the roof in this example, they must be cut carefully to take into account any variation in the height of the footers. This measurement will be done by using a transit, a laser level, or a water level to assess the difference in the height of the footers. The carpenter will then add or subtract length to the height of each post to compensate. Once the posts are cut, they can be raised into position, ensuring each is plumb (perfectly vertical) using a level.

Joists and Decking: The sizing for all the wooden parts of the deck is determined by calculating how long a distance must be spanned and the weight the span will carry. The timber that is parallel to the house and runs between the posts must be sized to be strong enough to carry all the weight between each pair of posts; the longer the span between posts, the larger this timber must be. Similarly, the floor joists that butt into this timber will need to be large enough to carry the weight over their length, and the decking will be sized so that it does not sag between the joists.

Fasteners: In our example, the deck will be fastened to the building using bolts, and held together using nails, while the decking itself will be screwed on. The carpenter has many fasteners to choose from with many different finishes. Each type of fastener has special characteristics that make it useful for certain tasks. Nails are typically sold by the pound and come in sizes from large 20d framing nails (often called 20 penny nails) to small 6d finish nails. Screws are also sold by the pound but are sized by length and by a number that can be converted, using a chart, to their diameter. Bolts are sold by diameter and length; as is the case with all fasteners, there are many different types among them, lag bolts, carriage bolts, and through bolts.

The Roof: The roof over the deck will be set at an angle so water runs off it and away from the house. The pitch of a roof is typically measured in “rise over run,” with the denominator of this fraction always given as 12. Thus, a roof that goes up four feet over a run of 12 feet is said to be a “4:12 roof.” The carpenter will use a special tool called a “speed square” that allows the direct conversion of roof pitch to angles and mark rafters for cutting.

Stairs: While stairs can be constructed to be more or less steep, a carpenter must keep in mind a basic mathematical relationship between tread length and riser height that will make a set of stairs comfortable to ascend. It turns out that because of the characteristic of the human gait, the steeper a stair, the less wide each tread should be. The formula that carpenters use is that for each stair, twice the rise plus the run should equal 24–26 inches.

Of course, once the carpenter is done with the project, there are still numerous other tasks to complete, including building railings and benches, as well as finishing and waterproofing the surfaces. If the homeowner were to want an outdoor grill area, with built-in cabinets, the carpenter would have a whole new set of challenges worthy of a cabinet maker and finish carpenter.

Bibliography

Gerhart, James. Mastering Math for the Buildings Trades. New York: McGraw-Hill, 2000.

Webster, Alfred P. Mathematics for Carpentry and the Construction Trades. 2nd ed. Upper Saddle River, NJ: Prentice Hall, 2001.