Part I: Problem Statement
The geometric problem that will be solved in this paper is about designing an umbrella. The analysis will cover different aspects, such as geometric design, shape, size, and amount of material to be used to construct the umbrella. With regard to geometry, round canopy umbrellas are very common. To construct such umbrellas, there is a need to have a certain number of sections before folding them. It is vital to note that the sections must be slopped. With regard to size, the radio cover ought to have a height of 60 cm and above for a normal person. In order to construct the umbrella, we need to make some assumptions with regard to the number of sections and height of the sections. In our analysis, it is assumed that our umbrella is octagonal in shape (when viewed from the top) and each section of the octagon has a height of 10 cm. This is illustrated in Figure 1.
Figure 1: Sectional view of an umbrella. Source Palason.
Assuming that our umbrella is made from the same material and color, we need to determine how much cloth is needed to construct our umbrella such that it looks like the one shown in Figure 2.
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Figure 2: Shape of our umbrella.
Part II: Solution
After the design, the next step involves providing a full solution and explanation for the problem. More specifically, we need to determine the amount of cloth needed to construct our umbrella. As stated earlier, our umbrella has eight sections that have an octagonal shape. Figure 3 shows a cross-section of the sections.
Figure 3: Cross-section
Using the Pythagoras theorem, we can determine the length of the hypotenuse.
Pythagoras theorem,
(Hahn, 2017)
Let the length of the hypotenuse be h,
Figure 4 shows the dimension of our umbrella’s top view,
Figure: Top view of our umbrella
As seen, the sections represent an isosceles triangle with a height of
This is illustrated in Figure 5. Let the base of our isosceles triangle be L.
Figure 5: Isosceles Triangle
To determine L, we have to determine the size of the angle a first
Using trigonometric rules, we can determine L.
(Sterling, 2014)
Next, we need to calculate the area of the isosceles triangle using the formula shown below
This means that each triangle has an area of 207.13 cm^2
But we have a total of 16 triangles (assuming the umbrella is made from the same material and color).
Therefore, the total area is
References
Hahn, R. (2017). The metaphysics of the Pythagoras theorem: Thales, Pythagoras, Engineering, diagrams and the construction of the cosmos out of right triangles. SUNNY Press.
Palason. (2020). Orange garden umbrella in 9 foot market style by Treasure Garden. [Online. Retrieved from: https://www.palason.ca/en/product/patio-umbrellas-/orange-garden-umbrella-in-9-foot-market-style-by-treasure-garden.html?SCATID=179&PID=10093&SECID=49&CATID=133 . Accessed January 22, 2019.
Sterling, M. (2014). Trigonometry for dummies. John Wiley & Sons.
