Trigonometry is perhaps one of the most widely used forms of mathematics in the fields of construction and engineering. This is because in order for whatever is being built to be functional and sturdy, precise math needs to be used to calculate exact values. If the blueprints of a project are inaccurate, then the flaws in the design could lead to a catastrophic failure later on, which is where trigonometry comes in. Trigonometry allows us to calculate exact values for triangles with minimal information about them to start. All shapes can be broken down into triangles, because they are the most basic polygon, and so therefore, trigonometry can be used to figure out measurements for more than just one type of shape. One particular field where trigonometry is widely used is that of constructing houses. Some of the many applications of trigonometry in this field include find the height of existing buildings with triangles, building trusses for roof support, and finding the desired roof pitch for a house. Also, although it does not relate to trigonometry directly, triangles are seen as the “strongest” shape, and are often used in construction, because they have the least amount of sides and angles a polygon can have.
One very basic application of trigonometry in the construction business is finding the height of existing buildings. This could be used in many different situations, like if a surveyor wanted to find the height of a house so that other houses could be built near it with the same height. To do this, the surveyor could measure the distance from where they are standing to the base of the structure they are looking for the height of. From that point, they could use surveying tools to find the angle of elevation from where they are standing to the top of the structure. By knowing these two measures, they could use the basic trigonometric identities to find the height of the building. In this particular case, if the height of the building was “x”, the distance from the surveyor to the building was “l”, and the angle of elevation to the top of the building was “a”, then the formula for the height of the building would be “x=l*tan(a)”. Since the tangent of a is “opposite/adjacent”, and x is the opposite side, you multiply both sides by the adjacent side value (l) to cancel it out on the left side of the equation, allowing you to solve for x.
Another very basic use of trigonometry is the use of triangles in construction. Since triangles are the simplest of polygons, they are the strongest. They have the least possible number of sides and angles, which makes them very sturdy and rigid, and allows them to hold a lot of weight. Since they only have three sides and angles, when they have weight put on them, it is distributed very evenly throughout the shape. When constructing houses, triangles are often used in foundations and roofs, to support some of the heavier parts of the house and distribute weight to the stronger parts of the house.
Another highly used application of trigonometry in construction is in roof trusses. Trusses are put in the attic of a house, underneath the roof, to support it and distribute the weight to the stronger parts of the foundation. They are usually composed entirely of triangles, because this allows them to be rigid and strong. Trusses such as the popular “W Truss”, “M Truss”, and “Scissors Truss” normally have standardized angles in the triangles they are composed of. For example, an M-Truss could be made out of two larger (30-60-90) scalene triangles making up the sections in the middle on the bottom, two smaller (30-60-90) scalene triangles making up the two parts on the outside on the bottom, and two (60-60-60) equilateral triangles making up the top and middle section. Knowing this, the width of the house, and the length of the roof, one can figure out, using trigonometry, how long each board needs to be in order for the truss to fit together correctly with the correct angles, so it has maximum stability. HowStuffWorks.com summarizes the use and advantages of trusses well by saying: “You can span a large distance with a truss and the truss transmits all of the weight to the exterior walls. Therefore, none of the interior walls are “load-bearing,” so they can go anywhere and are easily moved later.”
Lastly, trigonometry is used in constructing houses when figuring out roof pitch, and the lengths of boards needed to cover the entire roof. Depending on what type of house is being built, hundreds of different roof pitches can be used. The pitch determines how steep of an angle the roof comes down at. Once the desired pitch is known for the roof, the width and length of the house can be used in conjunction with the roof pitch angle to find out how long of boards are needed to cover the entire area. There is also a minimum roof pitch that houses must have in order to be able to drain rain, snow, leaves, and more off. Builders must keep this in mind when finding roof pitch, because if a roof has too low of a pitch, then additional drainage will need to be installed on top of the house to deal with weather issues. Roof pitch is written in the same way slope is. When finding roof pitch, it is written as a ratio of rise to run. An example, talking about a roof pitch of 7/12, taken from RoofGenius.com, is “So what does the 7/12 in the example … mean? The 7 means that the roof rises 7″ for every 12″ it runs.” In other words, for every 12 inches the roof goes horizontally, it goes up 7 inches vertically. A roof with a pitch of 7/12 has an angle of elevation of about 30 degrees. Roof pitch can be converted to a measure in degrees with the following formula: “Degrees=tan-1 (rise/run)”. This is because the tangent of the angle will equal “opposite/adjacent”. In this case, the rise is the opposite side and the run is the adjacent side. Since the tangent of the angle in degrees equals rise/run, then the inverse tangent of rise/run will equal the value of the angle in degrees. A generalization of how different roof slopes look by BuyersChoiceInspections.com is: “Flat Roof: 2/12, Low Slope: 2/12-4/12, Conventional Slope Roof: 4/12-9/12, Steep Slope: 9/12 and higher.
As you can see, trigonometry has many applications. The fields of construction and engineering themselves have hundreds of them. From the very basics of using triangles for their strength and sturdiness to calculating roof pitch to building trusses, there is a very wide range of techniques and math used in this field. Without trigonometry, a lot of the math used in construction would be much harder and time consuming, but thanks to it we can do it all rather quickly with little information.
Works Cited
Brain, Marshall. “How House Construction Works”. HowStuffWorks. 12/14/09 .
RoofGenius. “How to determine or calculate roof pitch or roof slope “. RoofGenius.com. 12/14/09 .
Buyer’s Choice Home Inspection. “Roof Slope”. Buyer’s Choice Home Inspection. 12/14/09 .
