I have been involved in landscaping for approximately 7 years. Most of the work that I have done focuses on commerical accounts with some very large residential. The accounts require yearly maintenance pertaining to keeping the property clean and manicured through out the year. With any landscaping of this scale, of course I have done irrigation or sprinkler work and installation, grading work, fencing, sod or new grass installation, among many types of other jobs that we might encounter.
In this essay, I want to focus on a particualr job that I recently encountered, and explore the math related to this job and how jobs of this type are often figured out by using good ole' five finger math or trial and error.
Recently I was sent to price and prepare a landscape timber tie wall. In the past I have had the opportunity to build many "tie" walls, and all of them were straight. This one on the other hand was not straight at all. It actually took the shape of a half-moon or half-circle. Since it was not likely that I would be able to buy lumber that took the shape of this half-circle, I figured that I had the perfect math problem in which to solve.
Problem: I needed to figure out how many cuts to make to carry the timbers around the half-circle where they would butt up together uniformly. once I figured thi out, I could then price the job accordingly and ensure the property manager of a quality job.
For an illustration of what the problem looked like, check out timber wall.
Solution 1: The first way I approached this problem was by trial and error. Even in the beginning, I did not like this method becaus each 6x6 timber is nearly 10-15 dollars depending on length. I started at the end of the circle and pulled a string to a point somewhere along the circle. Then measuring the length of the string, I started from the point on the circle and pulled the length of my string down to the next point on the cirle. I continued this until I reached the end and realized that each length was not the same once I came to the end. By adjusting the length of my string, I found how many points were needed which would be the points where the timbers would have to be cut. Once I had these points, we would lay a timber down overhanging the mark on the circle, lay another on top of it to the next point on the circle and trace the angle that the two made with each other with a pencil. This was not a bad idea, but since the tie wall was going to be six feet tall, I wanted something more exact.
Solution 2: More carefully searching for a mathematical way to approach this problem, I made a GSP sketch as shown earlier with a few variables. Let x be the radius of the circle by the reference point of the center of the circle. Let a and b denote where on the circle the timber wall will start and stop. Finally, let n be the timbers of a given length.
If we find the determined length of n by using a similar method as solution 1, we can now solve the problem and find out what angle needs to be cut to give the timber wall a uniform shape with no space between the angles.
If we look at one of the sections of our circle, we can drop the perpendicular bisector of the isosceles triangle, which will give us two rigth triangles of h, n/2, and x. Now we can solve for h in terms of x, and find the size of the angle by tan^-1(n/(4x^2-n^2)).
For a better illustration, click timber angles
One other thing that we can notice from this now that we have an equation is that we can change the length of our timbers and still derive the angle very simply by using the equation. This method seems to be much easier method or solution than by using the trial and error method in solution 1.
Another problem I wanted to discuss briefly is how to presicely measure the amount of sod it would take to cover a yard. When measuring sod, always take the measurement in square footage. If the area we are measuring is a square. or a figure that we know the common area of, this problem is not very difficult. But more difficult, would be to find the area of a square with a figure of either a flower bed or pinestraw island in the center of it. Even more difficult would be an irregular area of sod with several islands or odd shaped figures within it. I want to leave this problem to ponder by the reader, but from the timber wall problem, in the real-world, there is always two ways to solve a problem, an analytical way, as well as a "bulgarian peasant method"