Remarks on the Surface Area and Equality Conditions in Regular Forms Part II: Quadratic Prisms
Abstract
Following the same methodology and rules that were previously applied in Part I of this work, this part presents the remarks of the mathematical analysis for the regular quadratic right prisms. These include the rectangular and isosceles trapezoidal rooms. The first remark examines the effect of θ (or θ and β) on S. The second remark calculates the minimum total surface area (S_{Min}) in two cases, case of constant θ (or θ and β) and case of variable θ (or θ and/or β). The third remark calculates the two ratios R_{W} and R_{Wo}. The last remark studies the required conditions for the numerical equality between (Per–Ar), and (S–V).
Keywords
Trigonometry Algebra Differential equations Volume Area Total surface area Perimeter Regular polygons Right quadratic prisms Minimum total surface area Walls ratio Numerical equalityIntroduction
How the angle θ (or θ and β) affects S?
When S becomes minimum (S_{Min})?
What is the ratio between walls surface area S_{W} and S (S_{W}/S = R_{W})?
When Ar numerically equals Per? and,
When S numerically equals V?
Applying the same methodology and assumptions that were previously established in Part I, this part investigates the case of regular quadratic right prisms. The bases of such prisms could be either rectangular or isosceles trapezoidal, both will be considered in this part.
Notations
- Ar
: Room floor area (m^{2})
- h
: The diagonal of the rectangle or trapezoid (m)
- h_{0}
: The critical diagonal, the diagonal that fulfills (Per–Ar) equality (m)
- H_{R}
: Room height, the height of the prism (m)
- H_{Ro}
: The critical room height, the height that fulfills (S–V) equality (m)
- Per
: Perimeter (m)
- S
: Room total surface area (m^{2})
- S_{Min}
: The minimum total surface area (m^{2})
- S_{W}
: Walls total surface area (m^{2})
- R_{W}
: Walls ratio, S_{W}/S (Ratio)
- R_{Wo}
: The critical walls ratio, the ratio between walls total surface area and total surface area when S is minimum (S_{Min}) (Ratio)
- V
: Room volume (m^{3})
- ω_{o}
: The critical ratio, the ratio between H_{R} and r when S is minimum (S_{Min}) (Ratio)
The other terms will be illustrated in figures according to each case as required.
Rectangular Rooms
The Mathematical Relationships of Rectangular Prisms
Remark 1: Effect of θ on S
- Zone 1: This zone encloses between 0^{o} < θ ≤ 45^{o}, in this zone S is a decreasing function of θ. This zone can be also divided into two sub-zones:
Zone of rapid decay [a] (0^{o} < θ ≤ 7^{o}): where S loses more than 40 % of its maximum value.
Zone of slow decay [b] (7^{o} ≤ θ ≤ 45^{o}): θ increases rapidly in comparison with the reduction in S (in this zone, S loses about 10 % of its value at θ = 7^{o}).
Zone 2: in this zone S is an increasing function of θ. This zone (between 45^{o} ≤ θ < 90^{o}) can be also divided into two additional sub-zones [c] (up to θ ≤ 83^{o}), and [d]. Both zones are similar to the sub-zones [b] and [a] respectively.
Remark 2: the Minimum Total Surface Area, S_{Min}
Case of constant θ, where both Ar and H_{R} will be variables, or
Case of variable θ, where both Ar and H_{R} will be constants.
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. 4) and an increasing function of Ar (see Fig. 5), note that the location of the zones is reversed in Fig. 5. Thus, 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. This means that, unlike zone [a], an increase in H_{R} will increase S.
Case II, Variable θ, Constant Ar and H_{R}
This means that a room with a squared plan possesses the minimum perimeter among the other rectangular plans. Consequently, such a room has the minimum total surface area among the other rooms that have the same Ar and V but different θ. This result completely agrees with the findings of Sect. Remark 1 (see Fig. 2).
Remark 3: Walls Ratio R_{W}
Thus, the critical walls ratio R_{Wo} (see Sect. Notations) in rectangular right prisms is also constant for any θ and equals 2/3. This is similar to isosceles triangular right prisms (Elkhateeb 2014).
Remark 4: Case of Equality
This section calculates two cases of numerical equality in rectangular rooms. The first considers the equality between Per and Ar. The last considers the equality between S and V.
Case I, Equality of Per and Ar
Case II, Equality of S and V
Trapezoidal Rooms
The Mathematical Relationships of Regular Trapezoidal Prisms
Remark 1: Effect of θ on S
Remark 2: the Minimum Total Surface Area, S_{Min}
Case of constant θ and β, (variable Ar and H_{R})
- Case of variable θ and/or β, (constant Ar and H_{R}). This case can be divided into two additional sub-cases, these are:
Case of constant θ and variable β;
Case of variable θ and constant β.
Case I, Constant θ and β
Case I, Constant θ and β
By substitution in Eq. 47, h can be calculated. Knowing h, the other variables L, n and b can be calculated according to Eqs. 26, 28 and 31 respectively. Consequently the other parameters of the trapezoid can be calculated.
Example shows the effect of Ar and H_{R} on S in isosceles trapezoidal right prisms (constant θ and β, and variable Ar)
θ (°) | β (°) | V (m^{3}) | ω (ratio) | a (m) | b (m) | x (m) | h (m) | Ar (m^{2}) | H_{R} (m) | S (m^{2}) | ΔS (%) | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
15 | 20 | 4,500 | ω > ω_{o} | 0.78 | 7.25 | 21.52 | 20.22 | 28.44 | 202.21 | 22.25 | 2,585.00 | 10.0 |
0.67 | 7.63 | 22.65 | 21.29 | 29.94 | 224.05 | 20.08 | 2,519.66 | 7.2 | ||||
0.58 | 8.03 | 23.85 | 22.41 | 31.51 | 248.26 | 18.13 | 2,464.50 | 4.9 | ||||
0.49 | 8.45 | 25.10 | 23.59 | 33.17 | 275.08 | 16.36 | 2,419.74 | 3.0 | ||||
0.42 | 8.90 | 26.42 | 24.83 | 34.92 | 304.80 | 14.76 | 2,385.69 | 1.5 | ||||
0.36 | 9.37 | 27.81 | 26.14 | 36.75 | 337.73 | 13.32 | 2,362.74 | 0.5 | ||||
0.31 | 9.86 | 29.28 | 27.51 | 38.69 | 374.21 | 12.03 | 2,351.35 | 0.1 | ||||
ω_{o} | 0.29 | 10.09 | 29.95 | 28.15 | 39.58 | 391.69 | 11.49 | 2,350.14 | 0.0 | |||
ω < ω_{o} | 0.27 | 10.38 | 30.82 | 28.96 | 40.73 | 414.64 | 10.85 | 2,352.06 | 0.1 | |||
0.23 | 10.92 | 32.44 | 30.48 | 42.87 | 459.43 | 9.79 | 2,365.51 | 0.7 | ||||
0.20 | 11.50 | 34.15 | 32.09 | 45.13 | 509.07 | 8.84 | 2,392.44 | 1.8 | ||||
0.17 | 12.10 | 35.94 | 33.78 | 47.50 | 564.06 | 7.98 | 2,433.72 | 3.6 | ||||
0.14 | 12.74 | 37.84 | 35.55 | 50.00 | 625.00 | 7.20 | 2,490.32 | 6.0 | ||||
0.12 | 13.38 | 39.73 | 37.33 | 52.50 | 689.06 | 6.53 | 2,559.38 | 8.9 | ||||
0.11 | 14.05 | 41.72 | 39.20 | 55.13 | 759.69 | 5.92 | 2,644.39 | 12.5 | ||||
0.09 | 14.75 | 43.80 | 41.16 | 57.88 | 837.56 | 5.37 | 2,746.55 | 16.9 | ||||
0.08 | 15.49 | 45.99 | 43.22 | 60.78 | 923.41 | 4.87 | 2,867.23 | 22.0 | ||||
0.07 | 16.26 | 48.29 | 45.38 | 63.81 | 1,018.06 | 4.42 | 3,007.94 | 28.0 |
Case II, Variable θ and/or β
Case of constant θ and variable β
Case of variable θ and constant β
Case of constant θ and variable β
Case of variable θ and constant β
Equation 57 also reveals the results of the general case where both θ and β are variables. In such cases, S_{Min} exists when β reaches its maximum value (90^{o}). Thus, and according to Eq. 57, θ equals 45^{o}. This means that θ = ψ, or a squared shape. This result agrees with the findings of the rectangular prisms where a squared prism has the minimum S among other prisms that have the same Ar and V (see Sect. Case II, variable θ, constant Ar and H_{R}).
Remark 3: Walls Ratio R_{W}
The last equation indicates that R_{Wo}, similar to triangular and rectangular right prisms, is constant for any combination of θ and β and equals 2/3.
Remark 4: Case of Equality
Following the same methodology, this section calculates the two numerical equalities, (Per–Ar) and (S–V).
Case I, Equality of Per and Ar
Equation 63 reveals the condition under which this numerical equality exists. Again, and similar to ω_{o}, the numerical equality in this case is a function of both θ and β, for every combination of θ and β there is a specific h_{o} that fulfills this equality. Given the values of h_{o}, θ and β, the other parameters of a trapezoid: L, n, b, Per and Ar can be calculated from Eqs. 26, 28, 31, 36 and 45 respectively.
Case II, Equality of S and V
Similar to the triangular and rectangular rooms, the minus sign (-) in the denominator of Eq. 65 indicates that for every θ and β there is a minimum h under which this equality will never exist. Again, this occurs when H_{Ro} tends to ∞, i.e., when Ar equals Per according to Eq. 63. The relationship between Ar and H_{Ro} resembles the same relationship in the case of the rectangular right prisms (see Fig. 7).
Conclusions
Following the same methodology and rules that were applied previously in Part I, this part examines the cases of the regular quadratic right prisms. Such prisms include the rectangular and isosceles trapezoidal right prisms; both were considered in this part. The first remark examines the effect of θ on S. In the second remark, the minimum total surface area S_{Min} for the rooms under discussion was calculated in two cases, case of constant θ (or constant θ and β) and case of variable θ (or variable θ and/or β). In the first case, the critical ratio ω_{o} was calculated. Results showed that ω_{o} depends entirely on θ (or θ and β in case of isosceles trapezoidal right prisms). The values of ω_{o} were calculated and presented. In the second case, where θ (or θ and/or β in isosceles trapezoids) is variable, results showed that S_{Min}, in the case of rectangles, corresponds to θ = 45^{o}. In the case of trapezoids, results indicate that an isosceles trapezoidal right prism (with constant θ and variable β) will possess the minimum S when x → 0. In case of variable θ and constant β, the isosceles trapezoid that satisfies the condition of Eq. 57 possesses S_{Min}. The third remark calculates the ratio R_{W} (S_{W}/S). In rectangles, results showed that R_{W} reaches its minimum value when θ = 45^{o} whereas in trapezoids, it depends on the values of θ, β, h and H_{R}. Results also showed that the critical walls ratio R_{Wo} is constant for any θ (or θ and β in isosceles trapezoids) and is equal to 2/3. The last remark investigates the conditions for the numerical equality either between Per and Ar or S and V. In the first case, the critical diagonal h_{o} that fulfills Per–Ar equality was calculated. Results showed that h_{o} depends entirely on θ (or θ and β in isosceles trapezoids). In the second case, the critical room height H_{Ro} that fulfills S–V equality was calculated. Results also indicated that for every θ (or θ and β in isosceles trapezoids) there is a minimum h under which this equality will never exist; this corresponds to h_{o} (i.e. Ar = Per).
References
- Cremer, L., and H. Müller. 1978. Principles and applications of room acoustics. Translated by Theodore J. Schultz. London: Applied science PublishersGoogle Scholar
- Elkhateeb, A. 2014. Remarks on the surface area and equality conditions in regular forms, part I—triangular prisms. Nexus Network Journal, Architectural and Mathematics 16(1):219–232. doi:10.1007/s00004-014-0178-8