Remarks on the Surface Area and Equality Conditions in Regular Forms. Part I: Triangular Prisms
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Abstract
This work presents four mathematical remarks concluded from the mathematical analysis for the interrelationships between the dependent and independent variables that control the measures: perimeter, floor area, walls surface area and total surface area in the regular forms that have a given volume. Such forms include prismatic and pyramidal forms. The work consists of four parts, of which this first part presents the remarks of the isosceles triangular right prism. The first remark examines the effect of θ, the angle of the triangular base, on the total surface area. The second remark calculates the minimum total surface area in two cases, depending on whether angle θ is constant or variable. The third remark calculates the walls ratio and the critical walls ratio. The last remark studies the required conditions for the numerical equality in two cases, where the perimeter is equal to the area, and where the total surface area is equal to the volume.
Keywords
Trigonometry Algebra Differential equations Volume Area Total surface area Perimeter Regular polygons Right triangular prisms Minimum total surface area Walls ratio Numerical equalityIntroduction
Form is the visual appearance of a threedimensional object, and is often the main target in the architectural design. Form and space create the ambient world in which we live and experience our environment. According to Francis Ching, the form is established “… by the shapes and interrelationships of the planes that describe the boundaries of the volume” (Ching 2007, p. 28).
Mathematically, the perimeter (Per), floor area (Ar), total surface area (S) and volume (V) are four basic measures that can describe numerically any form. For our purposes here, a fifth value is also significant, θ, the angle of a triangular base. The interest in area and volume calculations has been started very early. It may go back to the period of the ancient Greeks (ca. eighth–sixth century BC) who calculated formulas to measure areas of simple geometrical shapes. Further, the ancient Greeks measured volumes according to their dry or liquid conditions suited, respectively to measuring grain and wine. In his “Elements”, Euclid presented many axioms and postulates related to the area of simple geometrical shapes. Nowadays, the methods and formulas to calculate Per, Ar, S and V for the common forms are almost available in every mathematical textbook [see, for example, (Ferguson and Piggott 1923; Bird 2003; Gieck and Gieck 2006)].
In modern advanced building analysis and design (such as room acoustics, artificial or day lighting and environmental control), areas and volumes are crucial measures. For example, in room acoustics, the ratio between Per and Ar determines the shape factor Sh _{ f }. Also, the ratio between V and S of a room determines its mean free path l and reverberation time T (Sabine 1993), both are basic measures in room acoustics (Elkhateeb 2012). In artificial lighting, room area is a main measure in the calculation of room cavity ratios that is a basic factor in lighting design (Grondzik et al. 2006). In environmental control, the areas of walls and/or roofs that face directly the sun control the amount of heat that transfers to the interior of a room according to the thermal connectivity and resistivity of their materials (Konya 2011). Thus, it is important for architects and practitioners in this field to be aware about the mathematical characteristics of these two measures (area and volume) and how they affect each other.
 The first includes all of the right prisms that have regular bases according to the definition of the term “regular” as used through this work (see Sect. “Definition of “Regular Forms” in this Work”). The plan is to cover the entire range of the regular basic shapes (from the triangle, to the circle). This group will be addressed in the first three parts of the work:

Part I: triangular prisms (this present paper);

Part II: quadratic prisms (including both: rectangle and trapezoid);

Part III: multisides prisms (from pentagon to circle).


The second includes all of the right regular pyramids either complete or incomplete (frustum of right pyramid). The base(s) has to be also a regular shape. These will be discussed in Part IV.
Definition of “Regular Forms” in this Work
In case of right prisms, the term “regular”, as used in this part and in the subsequent parts of the work, means that the bases of the prism have at least one axis of symmetry. Thus, in right triangular prisms, for example, only isosceles or equilateral triangles will be considered. In quadratic shapes only rectangular or symmetrical (isosceles) trapezoidal shapes will be considered. In the third dimension, the room is a right prism (i.e., all the sides of the prism are rectangles).
In the case of right pyramids, the term “regular” means that the base is a regular polygon (i.e., a multisides shape, from the equilateral triangle to the circle). In the third dimension, the apex of this pyramid is aligned directly above the center of its regular base.
The regular forms (or rooms, both terms could be used through the work) as described above have been chosen as the subject of this work because they are the most common in architectural applications. In addition, they can be grouped, organized and mathematically analyzed utilizing a consistent methodology. Although irregular rooms can be also studied using the same methodology, they have to be investigated separately according to the assumptions of each case but not as a group as the regular ones.
Problem Definition
Basic mathematical formulas to calculate the different measures (mainly, Per, Ar, S and V) of any regular form were established many years ago and are wellknown. Nevertheless, to my knowledge, there is no advanced study that analyzes the interrelationships between these variables (dependent and independent). Consequently, a discussion of the way they affect each other is still lacking.
Objectives

How the angle θ (or θ and β) affects S;

When S becomes minimum (S_{Min});

The ratio between walls surface area S_{W} and S (S_{W}/S = R_{W});

When Ar numerically equals Per;

When S is numerically equal to V.
Methodology

Case of constant θ with variable Ar, and H_{R};

Case of variable θ with constant Ar and H_{R}.
For both cases, the derived functions were used to examine the effect of θ (or θ and β) on S and to calculate S_{Min} for the room under discussion. Finally, the conditions for the equality were calculated utilizing the rules of algebra and trigonometry based on the derived functions.
The Mathematical Relationships of Regular Triangular Prisms
Remark 1: Effect of θ on S
 Zone 1: This zone encloses between 0^{o} < θ ≤ 60^{o}, in this zone S is a decreasing function of θ. This zone can be also divided into two subzones:

Zone of rapid decay (a) (0^{o} < θ ≤ 15^{o}): where S loses about 60 % of its maximum value.

Zone of slow decay (b) (15^{o} ≤ θ ≤ 60^{o}): θ increases rapidly in comparison with the reduction in S (in this zone, S loses about 22 % of its value at θ = 15^{o}).


Zone 2: in this zone S is an increasing function of θ. This zone (between 60^{o} ≤ θ < 90^{o}) can be also divided into two additional subzones (c) (up to θ ≤ 85^{o}), and (d). Both zones are almost identical to the subzones (b) and (a), respectively.
It can also be concluded from Fig. 3 that the variation in S corresponding to θ in the range between 45^{o} and 60^{o} is limited and can be ignored (Elkhateeb 2012). Nevertheless, beyond this range (whether θ ≥ 60^{o} or θ ≤ 45^{o}) this variation is obvious and must be considered upon deciding the dimensions and setups of a room. One can argue that values of θ outside this range (45^{o}–60^{o}) are not common in architectural applications. However, we cannot rely on that, as everything is possible in architecture. More discussion about the relationship between θ and S is presented in “Case II, Variable θ, Constant Ar and H_{R} ”.
Remark 2: the Minimum Total Surface Area, S_{Min}

Constant, in this case both Ar and H_{R} will be the variables, or

Variable, in this case both Ar and H_{R} will be constants.
Thus, Remark 2 will be divided into two subremarks in order to discuss both cases.
Case I, Constant θ, Variable Ar and H_{R}

Zone (a): where ω < ω _{ o }. In this zone, S is a decreasing function of H_{R} (see Fig. 5) and an increasing function of Ar (see Fig. 6), note that the location of the zones is reversed in this last figure. This means that any increase in room height will decrease its total surface area.

Zone (b): where ω > ω _{ o }. In this zone, S is an increasing function of H_{R} and a decreasing function of Ar (see Figs. 5 and 6). This means that an increase in H_{R} will increase S. This is the opposite of what happens in zone (a).
Example shows the effect of Ar and H_{R} on S (constant θ and variable Ar)
θ ^{o}  V (m^{3})  ω (ratio)  h (m)  H_{R} (m)  Ar (m^{2})  Per (m)  S (m^{2})  ΔS (%)  

45 ^{ o }  4,000  ω > ω _{ o }  1.47  13.97  20.50  195.12  67.45  1,772.90  3.43 
1.41  14.14  20.00  200.00  68.28  1,765.69  3.01  
1.36  14.32  19.50  205.13  69.15  1,758.76  2.61  
1.31  14.51  19.00  210.53  70.06  1,752.16  2.22  
1.26  14.70  18.50  216.22  71.00  1,745.91  1.86  
1.21  14.91  18.00  222.22  71.98  1,740.05  1.52  
1.16  15.12  17.50  228.57  73.00  1,734.62  1.20  
1.11  15.34  17.00  235.29  74.06  1,729.69  0.91  
ω _{ o }  0.83  16.90  14.00  285.67  81.61  1,714.04  0.00  
ω < ω _{ o }  0.36  22.36  8.00  500.00  107.97  1,863.74  8.73  
0.32  23.09  7.50  533.33  111.51  1,902.97  11.02  
0.29  23.90  7.00  571.43  115.42  1,950.81  13.81  
0.26  24.81  6.50  615.38  119.78  2,009.33  17.23  
0.23  25.82  6.00  666.67  124.67  2,081.35  21.43  
0.18  28.28  5.00  800.00  136.57  2,282.84  33.18  
0.15  29.81  4.50  888.89  143.96  2,425.58  41.51  
0.13  31.62  4.00  1,000.00  152.69  2,610.75  52.32 
Case II, Variable θ, Constant Ar and H_{R}
This result comes in a complete agreement with the findings of Remark 1 (see Fig. 3). It also agrees with the mathematical fact that the isosceles triangle has the minimum perimeter among the other triangles (Alsina and Nelsen 2009). Further, the equilateral triangle (θ = 60^{o}) possesses the absolute minimum perimeter, consequently the minimum total surface area among the other rooms that have the same Ar and V but different θ.
Remark 3: Walls Ratio R_{W}
The relationship between R_{W} and θ resembles the relationship between S and θ (see Fig. 3), thus R_{W} reaches its minimum value when θ = 60^{o}. In the zone where θ < 60^{o}, R_{W} is a decreasing function of θ. In the zone where θ > 60^{o}, R_{W} is an increasing function of θ.
This means that R _{ Wo } is constant for any θ (0^{o} < θ < 90^{o}) and is equal to 2/3.
Remark 4: Case of Numerical Equality
For the room under discussion, two cases of numerical equality will be examined. The first considers the numerical equality between the perimeter Per and the floor area Ar. The last considers the numerical equality between the total surface area S and the volume V.
Case I: Equality of Per and Ar
Case II: Equality of S and V
Conclusions
This work has examined the interrelationships between the dependent and independent variables that control the values of the measures Per, Ar, S and V and how these variables affect each other in the case of regular triangular right prisms. Under the conditions assumed for this work, four remarks were concluded. In the first, the effect of θ on S was investigated. In the second remark, the minimum total surface area S_{Min} for the room under discussion was calculated in two cases, case of constant θ and case of variable θ. In the first case, new variable (ω = H_{R}/h) was introduced. For every θ there is a specific ω (called ω _{ o }) that results S_{Min}. Results showed that ω _{ o } depends entirely on θ. The values of ω _{ o } in the range 20^{o} ≤ θ ≤ 80^{o} were calculated and presented. In the second case, where θ is variable, results showed that S_{Min} corresponds to θ = 60^{o}. The third remark calculates walls ratio R_{W}, results showed that R_{W} reaches its minimum value when θ = 60^{o}. In case of the isosceles triangular right prism that has dimensions fulfill ω _{ o }, R_{W} was called R _{ Wo }. Results showed that R _{ Wo } is constant (=2/3) regardless the value of θ.
The last remark investigates the conditions for the numerical equality either between Per and Ar or S and V. In the first case, another variable (h _{ o }) was introduced. Results showed that h _{ o } depends also on θ. For every θ there is a specific h _{ o } that fulfills the numerical equality between Per and Ar. The values of h _{ o } in the range 20^{o} ≤ θ ≤ 80^{o} were calculated and presented. In the second case, the condition for the numerical equality between S and V was calculated. Results showed that for every θ and Ar, there is a specific H_{R} (called H _{ Ro }) that fulfills this equality. Results also showed that for every θ there is a minimum h under which this equality will never exist. This corresponds to h _{ o } (i.e., Ar = Per).
Notes
Acknowledgments
The author would like to express his gratitude and sincere appreciation to all colleagues who make this work possible. In particular, thanks to Prof. Dr. Morad Abdel Kader, Dr. Esraa Elkhateeb and Dr. Ahmed Zakareia for their valuable discussion and help during the analysis.
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