Nexus Network Journal

, Volume 16, Issue 3, pp 795–811

Remarks on the Surface Area and Equality Conditions in Regular Forms Part IV: Pyramidal Forms

Didactics

DOI: 10.1007/s00004-014-0204-x

Cite this article as:
Elkhateeb, A.A. & Elkhateeb, E.A. Nexus Netw J (2014) 16: 795. doi:10.1007/s00004-014-0204-x
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Abstract

Following the same methodology and rules that were applied in the previous Parts I–III of this work, Part IV presents the mathematical remarks on the right regular pyramidal forms, either complete or truncated (a frustum of pyramid). The first remark examines the effects of θ and β on S. The second remark calculates the minimum total surface area (SMin) in two cases, case of constant θ (variable β) and case of variable θ (constant β). The third remark calculates walls ratio RW and the critical walls ratio RWo. The last remark calculates the numerical equality between S and V. In conclusion, the importance of the findings of the entire work (Parts I–IV) in advanced building analysis is highlighted and discussed.

Keywords

Acoustics Pyramids Pyramidal forms Surface area Walls ratio Building analysis Minimum total surface area Numerical equality 

Introduction

Pyramidal forms are some of the oldest forms that humans have ever known. Historically, pyramids were constructed in different compositions (complete, truncated and stepped) in many places of ancient world, such as Egypt, Mexico and Latin America. The step pyramid of Djoser (Zoser, in the Saqqara necropolis) may be the first of this type (built around 2800 BC). In the realm of modern architecture, the pyramidal form still attracts architects. There are several examples for modern pyramidal buildings in different places around the world, for example the Unknown Soldier Memorial in Cairo, by Sami Rafi (1973), Louvre pyramid in Paris, by I. M. Pei (1989) and Luxor Hotel, Las Vegas, by Veldon Simpson (1993).

This Part IV applies the same methodology and assumptions that were established in the previous Parts I–III (Elkhateeb 2014a, b; Elkhateeb and Elkhateeb 2014) of this work to investigate the case of the right regular pyramid (either complete or truncated) so as to answer the following questions:
  • How do the angles θ and β affect S?

  • When does S become minimum (SMin)?

  • What is the ratio between walls total surface area SW and S (SW/S = RW)?

  • When S equals V?

Notations

In this work, the following terms mean:
Ar

Base area, area of the lower base (truncated pyramid) (m2)

Ars

Area of one side face of the pyramid (m2)

ArU

Area of the upper base (truncated pyramid) (m2)

βo

The critical dihedral angle, the angle at which S of a complete right regular pyramid becomes minimum

h

The altitude of the triangle (m)

hs

Slant height (m)

HR

The altitude of the pyramid (m)

HRo

The critical room height, the height that fulfills (S − V) equality (m)

n

Number of sides

Per

Perimeter (m)

S

Room total surface area (m2)

SMin

The minimum total surface area (m2)

SW

Walls total surface area (m2)

r

Radius (m)

ro

The critical radius, the radius that fulfills (Per − Ar) equality (m)

RW

Walls ratio, SW/S (ratio)

RWo

The critical walls ratio, the ratio between walls total surface area and total surface area when S is minimum (SMin) (ratio)

V

Room volume (m3)

ωo

The critical ratio, the ratio between HR and r when S is minimum (SMin) (ratio)

The other terms will be illustrated in figures according to each case as required.

Right Regular Pyramid

A right regular pyramid is a pyramid that has its apex aligned directly above the center of its regular base. This base can be any regular polygon (a multi-sided shape) from the equilateral triangle (n = 3) to the circle (n = ∞) in which:
  • All sides are congruent.

  • All angles are congruent (thus, the angle ψ is constant and equal to 360/n).

Through this part, it is assumed that V, θ and β are the independent variables whereas Ar, Per, and S are the dependent ones. Figure 1 illustrates the different variables: θ, ψ, h, r, L, β, HR and hs in a right regular pyramid.
Fig. 1

Right regular pyramids, the different variables: aleft a regular multi-sided base; bright a pyramidal form. Image: author

The Mathematical Relationships of Right Regular Pyramids

In Part III of this work, the main mathematical functions between θ, L, h, r, Per and Ar have been driven. These functions are also applied in this part as long as the base of the pyramid is regular as defined and assumed. While this section will not repeat those functions of Part III, it will derive additional mathematical functions among r, S, β and V (see Fig. 1). These new functions will be utilized later to determine SMin and calculate the equality conditions.

A right regular pyramid can be fully identified when its volume V, number of sides n (thus θ) and the dihedral angle β are known. Given these variables, it can be proved that (see Part III):
$$ r = \sqrt[3]{{\frac{3V}{{n\sin \theta { \cos }^{2} \theta { \tan }\beta }}}}. $$
(1)
This equation yields the radius of a circle that contains the regular base of a pyramid knowing n, V and β. In addition, it also relates r to the dihedral angle β for a given V. Given r, both HR and Ar can be calculated as:
$$ H_{R} = r\cos \theta \tan \beta $$
(2)
and (Gieck and Gieck 2006)
$$ Ar = \frac{3V}{{H_{R} }} . $$
(3)
From Eq. 3, HR can be also calculated as:
$$ H_{R} = \frac{3V}{{n r^{2} { \sin } \theta { \cos }\theta }} . $$
(4)
In such pyramid, there are two possibilities for modifying its dimensions assuming that V is constant:
  • In the first, β and Ar are variables whereas n (thus θ) is constant.

  • In the second, β is constant whereas n and Ar (i.e., r) are variables.

As can be concluded from the previous options, Ar is always variable as long as both V and β were determined. From the first principles, the slant height hS can be calculated as:
$$ h_{S} = \frac{{H_{R} }}{\sin \beta } . $$
(5)
The area of one face of this pyramid ArS can be calculated from:
$$ Ar_{S} = \frac{{L H_{R} }}{{2 { \sin }\beta }} . $$
(6)
The total surface area of a right regular pyramid can be calculated as:
$$ S = \frac{{nrH_{R} \sin \theta }}{\sin \beta } + nr^{2} { \sin } \theta { \cos } \theta . $$
(7)
By substitution for HR according to Eq. 4, Eq. 7 can be rewritten as:
$$ S = \frac{3V}{{r { \cos } \theta { \sin }\beta }} + nr^{2} { \sin } \theta { \cos } \theta . $$
(8)
If r is replaced by its equivalent value according to Eq. 1, Eq. 8 can be rewritten as:
$$ S = \frac{3V}{{\left( {\sqrt[3]{{\frac{3V}{{n\sin \theta \cos^{2} \theta \tan \beta }}}}} \right) \times \cos \theta \sin \beta }} + \left( {\frac{3V}{{n\sin \theta \cos^{2} \theta \tan \beta }}} \right)^{\frac{2}{3}} \times n\sin \theta \cos \theta . $$
(9)
Given that n = 180/θ, Eq. 9 will be:
$$ S = \frac{3V}{{\left( {\sqrt[3]{{\frac{3V}{{(180/\theta )\sin \theta \cos^{2} \theta \tan \beta }}}}} \right) \times \cos \theta \sin \beta }} + \left( {\frac{3V}{{(180/\theta )\sin \theta \cos^{2} \theta \tan \beta }}} \right)^{\frac{2}{3}} \times \left( {\frac{180}{\theta }} \right)\sin \theta \cos \theta . $$
(10)

Remark 1: Effects of θ and β on S

The effect of θ on S in the right regular pyramid can be concluded from Eq. 7 when β is constant (Fig. 2). This effect resembles the case of the regular multi-sided right prisms (see Part III). In summary, S is an increasing function of θ (thus a decreasing function of n). This means that S of a right regular triangular pyramid (n = 3) is larger than a right cone (n = ∞) provided that both have the same V (Elkhateeb 2014b).
Fig. 2

Effect of θ on S according to Eq. 7 (different values of β)

Similarly, the effect of β on S can be also concluded from Eq. 7 when θ is constant. Figure 3 illustrates this case graphically. It is clear from this figure that the function is similar to the case of the regular right prisms (see for example Part III) where the function changes its direction at a certain angle. This angle will be called βo (see “Case I: Variable β and Constant θ”). βo splits the function into two main zones:
Fig. 3

Effect of β on S according to Eq. 7 (case of n = 3)

  • Zone [a]: where 0° < β < βo. In this zone S is a decreasing function of β.

  • Zone [b]: where βo < β < 90°. In this zone S is an increasing function of β.

Remark 2: The Minimum Total Surface Area, SMin

Following the same approach that was applied previously in the earlier parts of this work and as mentioned in “The Mathematical Relationships of Right Regular Pyramids”, two cases will be considered:
  • Case of variable β and constant θ;

  • Case of constant β and variable θ.

Case I: Variable β and Constant θ

In this case, among the different right regular pyramids that have the same θ and V, SMin occurs when the first derivative of Eq. 9 equals zero
$$ \frac{ds}{d\beta } = \frac{1}{3}\frac{{3^{2/3 } V^{2} (1 + { \tan }^{2} \beta )}}{{\left( {\frac{V}{{n {\text{sin }}\theta {\text{cos }}^{2} \theta \tan \beta }}} \right)^{4/3} {\text{cos }}^{3} \theta { \sin } \theta { \sin }\beta { \tan }^{2} \beta }} - \frac{{3^{2/3 } V {\text{cos }}\beta }}{{\left( {\frac{V}{{n { \sin } \theta {\text{cos }}^{2} \theta { \tan }\beta }}} \right)^{1/3} { \cos } \theta { \sin }^{2} \beta }} - \frac{2}{3} \frac{{3^{2/3 } V \left( {1 + { \tan }^{2} \beta } \right)}}{{\left( {\frac{V}{{n { \sin } \theta {\text{cos }}^{2} \theta {\text{tan }}\beta }}} \right)^{1/3} {\text{cos }} \theta { \tan }^{2} \beta }} = 0 . $$
(11)
The solution of Eq. 11 yields the value of βo, the dihedral angle at which S reaches its minimum value. As can be seen, this last equation is very complex to solve analytically in order to get the value of βo. Possible solutions for such problem are numerical calculations or graphical representation for the function dS/dβ. In our case, the program Maple® for symbolic calculations was utilized to get the value of βo. The calculations lead to:
$$ \beta_{o} = arctan (2\sqrt 2 ),\quad i.e. \, \beta_{o} = 70.528779^{^\circ } . $$
(12)
Thus, a right regular pyramid that has a dihedral angle equal to βo possesses the minimum total surface area among others that have the same θ and V. In the right regular pyramids, the critical ratio ωo can be calculated as:
$$ \omega_{o} = 2\sqrt {2 } { \cos } \theta . $$
(13)
It can be concluded from Eq. 13 that ωo also depends on θ and is an increasing function of θ (thus a decreasing function of n). Table 1 lists the values of ωo for the most common right regular pyramids according to their bases. ωo can be utilized as an alternate way to calculate SMin for a right regular pyramid when its V, θ and β are known by applying Eq. 1 to calculate r, then Eq. 13 to get HR. The other variables can be calculated using the appropriate formulas (see “The Mathematical Relationships of Right Regular Pyramids” here and Part III (Elkhateeb 2014b).
Table 1

Values of ωo for the common right regular pyramids according to their bases

θo

60

45

36

30

22.5

0

Shape of base

Triangle

Square

Pentagon

Hexagon

Octagon

Circle

ωo

1.4142

2.00

2.288

2.449

2.613

2.828

Similar to the cases of triangular and quadratic right prisms, the two relationships (HR–S) and (Ar–S) depend on βo, which divides the functions into two zones (see Figs. 4, 5):
Fig. 4

The relationship of HR to S (case of triangle, n = 3)

Fig. 5

The relationship of Ar to S (case of pentagon, n = 5)

  • Zone [a]: where β < βo. In this zone, S is a decreasing function of HR (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 pyramid height will decrease its total surface area.

  • Zone [b]: where β > βo. In this zone, S is an increasing function of HR and a decreasing function of Ar. This means that an increase in HR will increase S.

Case II: Constant β and Variable θ

In this case, among the different right regular pyramids that have the same β and V, SMin occurs when the first derivative of Eq. 10 equals zero. Thus,
$$ \begin{aligned} & \frac{ds}{d\theta } \\ & \quad = - \frac{1}{3}\frac{{3^{2/3} V^{2} \left( {\frac{1}{{180 { \sin } \theta { \cos }^{ 2} \theta { \tan } \beta }} - \frac{\theta }{{180 { \sin } ^{2} \theta {\text{cos}} \theta {\text{tan }}\beta }} + \frac{2\theta }{{180 { \cos }^{ 3} \theta {\text{tan }}\beta }}} \right)}}{{\left( {\frac{V \theta }{{180 { \sin } \theta {\text{cos}}^{2} \theta {\text{tan}} \beta }}} \right)^{4/3} { \cos }\theta { \sin } \beta }} \\ & \quad \quad + \frac{{3^{2/3} V {\text{sin }} \theta }}{{\left( {\frac{V \theta }{{180 { \sin } \theta { \cos }^{2} \theta { \tan } \beta }}} \right)^{1/3} { \cos }^{2} \theta {\text{sin }} \beta }} \\ & \quad \quad + \frac{2}{3}\frac{{3^{2/3} \times 180 V { \cos }\theta {\text{sin }} \theta \left( {\frac{1}{{180 {\text{sin }}\theta { \cos }^{2} \theta { \tan }\beta }} - \frac{\theta }{{180 { \sin } ^{2} \theta {\text{cos }}\theta {\text{tan }}\beta }} + \frac{2\theta }{{180 { \cos }^{3} \theta {\text{tan}} \beta }}} \right)}}{{\left( {\frac{V \theta }{{180 { \sin } \theta { \cos }^{ 2} \theta { \tan } \beta }}} \right)^{4/3} \theta }} \\ & \quad \quad - \frac{1}{{\theta^{2 } }}\left( {\frac{3 V \theta }{{180 {\text{sin}} \theta { \cos }^{2} \theta { \tan } \beta }}} \right)^{2/3} 180 {\text{sin }}\theta { \cos } \theta + \frac{180}{\theta }\left( {\frac{3 V \theta }{{180 { \sin } \theta { \cos }^{2} \theta { \tan } \beta }}} \right)\left( {{ \cos }^{2} \theta - {\text{sin }}^{2} \theta } \right) = 0. \\ \end{aligned} $$
(14)

Again, due to the complexity of this last equation, the program Maple® for symbolic calculations was utilized to get the value of θ at which S becomes minimum. The calculations indicate that SMin occurs when θ → 0.

Thus, among the different right regular pyramids, a cone has the minimum total surface area. This conclusion completely agrees with the numerical solution presented in Fig. 2 and discussed in “Remark 1: Effects of θ and β on S”.

Remark 3: Walls Ratio RW

In right regular pyramids, RW can be mathematically defined as:
$$ R_{W} = \frac{{n \times Ar_{S} }}{{Ar + n \times Ar_{S} }} . $$
(15)
By substitution for ArS according to Eq. 6 and Ar (see Part III), Eq. 15 can be rewritten as:
$$ R_{W} = \frac{{H_{R} }}{{H_{R} + r { \cos } \theta { \sin } \beta }} . $$
(16)
If HR is replaced by its equivalent value according to Eq. 2, then Eq. 16 will be:
$$ R_{W} = \frac{{{ \tan } \beta }}{{{ \tan }\beta + { \sin } \beta }} . $$
(17)
According to Eq. 17, RW depends completely on β regardless the values of V and θ. For example, RW = 0.586 for β = 45o. Figure 6 represents the relationship β–RW, it is clear that RW is an increasing function of β. According to Eq. 17, it can be concluded that RWo = 0.75.
Fig. 6

The ratio RW as a function of β

Remark 4: Case of Numerical Equality

In Part III (Elkhateeb 2014b), the numerical equality between Ar and Per was calculated. As the same condition applies here, it is not repeated again. Thus, this section considers only the numerical equality between S and V in right regular pyramids.

Equality of S and V

According to Part III, Eqs. 2 and 7, this numerical equality occurs when:
$$ \frac{{nrH_{R} { \sin } \theta }}{\sin \beta } + nr^{2} { \sin } \theta { \cos } \theta = \frac{1}{3}H_{R} nr^{2} { \sin } \theta { \cos } \theta . $$
(18)
By applying the principles of algebra, Eq. 18 will be:
$$ H_{Ro} = \frac{{3r\cos \theta { \sin } \beta }}{{r\cos \theta { \sin } \beta - 3}}. $$
(19)
Thus, in right regular pyramids with given θ, β and Ar, the numerical equality between S and V occurs only if Eq. 19 has been satisfied. This can be calculated in the following sequence:
  • Determine θ (or n), β and Ar;

  • Apply Eq. 4 (Part III) to get r;

  • Substitute in Eq. 19 to get the critical room height HRo (see “Notations”) that fulfills this equality.

The relationship Ar–HRo resembles the previous cases studied in this work. The minus sign \( ( - ) \) in the denominator of Eq. 19 indicates the limit under which this equality will never exist. Mathematically, this equality will never exist if:
$$ r\cos \theta { \sin }\beta \le 3. $$
(20)
In the special case where n → ∞ (i.e., a cone), θ → 0. As cos 0 = 1, thus, Eq. 19 will be:
$$ H_{Ro} = \frac{{3r { \sin } \beta }}{{r { \sin } \beta - 3}} . $$
(21)

Truncated Right Regular Pyramid

The truncated right regular pyramid is a portion of a right regular pyramid included between two parallel bases. In such a pyramid:
  • The slant height hS is the altitude of a side face.

  • The lateral edges are equal, and the side faces are equal isosceles trapezoids.

  • The two bases are similar parallel regular polygons.

In addition to the variables V, θ and β which identified the complete right regular pyramid, the height HR (see Fig. 7) must be also identified in case of a frustum of a pyramid. Thus, in the following section, it is assumed that V, θ, β and HR are the independent variables whereas Ar, Per, and S are the dependent ones. Figure 7 illustrates the variables: θ, β, r, rU, L, LU, HR, δ and hS.
Fig. 7

Truncated right regular pyramid, different variables, see also Fig. 1 for the other variables

The Mathematical Relationships of Truncated Right Regular Pyramids

Beside the main mathematical relationships previously listed in Part III and “The Mathematical Relationships of Right Regular Pyramids” of this part, this section lists additional mathematical formulas regarding the truncated right regular pyramid. From the first principles, it can be concluded that:
$$ \delta = \frac{{H_{R} }}{{{\text{tan }}\beta }} $$
(22)
and
$$ h_{U} = h - \delta . $$
(23)
By replacing h, hU and δ with their equivalent values (see Part III, Elkhateeb 2014b), then Eq. 23 will be:
$$ r_{U} = r - \frac{{H_{R} }}{{{ \cos } \theta { \tan } \beta }} . $$
(24)
From the first principles again, the total surface area S of a truncated right regular pyramid can be calculated as:
$$ S = n h_{S} \left( {\frac{{L + L_{U} }}{2}} \right) + Ar + Ar_{U} $$
(25)
and the volume of such pyramid can be calculated from
$$ V = \frac{1}{3} H_{R} \left( {Ar + Ar_{U} + \sqrt {Ar.Ar_{U} } } \right) $$
(26)
(Gieck and Gieck 2006).
If Ar and ArU are replaced by their equivalent values (see Part III, Elkhateeb 2014b), then Eq. 26 will be:
$$ V = \frac{1}{3} n H_{R} { \sin } \theta {\text{cos }}\theta \left( {r^{2} + r_{U}^{2} + r \cdot r_{U} } \right). $$
(27)
By substitution for rU according to Eq. 24, Eq. 27 will be:
$$ V = \frac{1}{3} H_{R} { \sin } \theta { \cos } \theta \left[ {r^{2} + \left( {r - \frac{{H_{R} }}{{{ \cos } \theta { \tan }\beta }}} \right)^{2} + r.\left( {r - \frac{{H_{R} }}{{{ \cos }\theta { \tan } \beta }}} \right)} \right]. $$
(28)
This leads to a quadratic equation with one unknown, that is, r:
$$ 3 r^{2} - \frac{{3 r H_{R} }}{{{ \cos }\theta { \tan } \beta }} + \left[ {\left( {\frac{{H_{R}^{2} }}{{{ \cos }^{2} \theta { \tan }^{2} \beta }}} \right) - \left( {\frac{3 V}{{n H_{R} { \sin } \theta { \cos } \theta }}} \right)} \right] = 0 . $$
(29)
This last equation yields r for a truncated right regular pyramid given its independent variables V, θ, β and HR. Given r, the other variables of such pyramid can be calculated. As a quadratic equation, its three constants a, b and c can be calculated as:
$$ a = 3; $$
(30)
$$ b = - \frac{{3H_{R} }}{{{\text{cos}}\theta {\text{tan }}\beta }} ; $$
(31)
$$ c = \frac{{H_{R}^{2} }}{{{ \cos }^{2} \theta { \tan }^{2} \beta }} - \frac{3 V}{{n H_{R} { \sin } \theta { \cos } \theta }} . $$
(32)
In such a pyramid, there are three possibilities for modifying its dimensions, assuming that V is constant:
  1. 1.

    β and Ar are variables whereas n (thus θ) is constant.

     
  2. 2.

    β is constant whereas n and Ar (i.e., r) are variables.

     
  3. 3.

    HR is variable whereas both β and θ are constants.

     

Again, in all cases Ar is variable as long as V, β and HR were determined.

Remark 1: Effects of θ, β and HR on S

In a right regular truncated pyramid that has a given V, the numerical solutions for Eqs. 25 and 29 indicate that:
  • S is an increasing function of θ when HR and β are constants. Thus SMin occurs when θ → 0 (i.e., a cone). Figure 8 represents θ–S relationship for different cases of β.
    Fig. 8

    The relationship of θ to S, different cases of β

  • S is a decreasing function of β when HR and θ are constants. Thus SMin occurs when β → 90 (i.e., a prism). Figure 9 represents β–S relationship for different cases of (n).
    Fig. 9

    The relationship of β to S, different cases of (n)

  • S is a decreasing function of HR when θ and β are constants, thus SMin occurs when HR is maximum. Figure 10 represents HR–S relationship for different cases of θ and β.
    Fig. 10

    The relationship of HR to S, different cases of θ and β

Remark 2: Walls Ratio RW

In a right regular truncated pyramid, RW can be mathematically calculated as:
$$ R_{W} = \frac{{n h_{S} \left( {\frac{{L + L_{U} }}{2}} \right)}}{{n h_{S} \left( {\frac{{L + L_{U} }}{2}} \right) + Ar + Ar_{U} }} . $$
(33)
Given the equivalent values of hS, L, LU and Ar (see Part III), Eq. 33 can be rewritten as:
$$ R_{W} = \frac{{H_{R} {\text{sin }}\theta (r + r_{U} )}}{{H_{R} {\text{sin }}\theta \left( {r + r_{U} } \right) + { \cos } \theta { \sin } \theta { \sin } \beta (r^{2} + r_{U}^{2} )}} . $$
(34)
If HR was replaced by its equivalent value according to Eqs. 22 and 23, then Eq. 34 will be:
$$ R_{W} = \frac{{r^{2} - r_{U}^{2} }}{{(r^{2} - r_{U}^{2} ) + { \cos } \beta (r^{2} + r_{U}^{2} )}} . $$
(35)

Similar to the case of the complete pyramid, Eq. 35 says that RW is independent on both V and θ, but it is a function of r, rU and β. It is clear that RWo will never exist as long as the three functions (θ–S, β–S and HR–S) remain constant in direction.

Remark 3: Case of Numerical Equality

As mentioned above, the condition for the numerical equality between Ar and Per is the same as the case of regular multi-sided right prisms. Thus, the following section considers only the case of numerical equality between S and V.

Equality of S and V

According to Eqs. 25 and 27, this numerical equality occurs when:
$$ n h_{S} \left( {\frac{{L + L_{U} }}{2}} \right) + Ar + Ar_{U} = \frac{1}{3} n H_{Ro} { \sin } \theta {\text{cos}} \theta \left( {r^{2} + r_{U}^{2} + r.r_{U} } \right). $$
(36)
By replacing hS, L, LU, Ar and ArU with their equivalent values, Eq. 36 will be:
$$ H_{Ro} { \cos } \theta { \sin } \beta \left( {r^{2} + r_{U}^{2} + r.r_{U} } \right) - 3H_{Ro} \left( {r + r_{U} } \right) - 3 {\text{cos}} \theta { \sin } \beta \left( {r^{2} + r_{U}^{2} } \right) = 0 . $$
(37)
By substitution for rU according to Eq. 24, Eq. 37 can be rewritten as:
$$ H_{Ro}^{3} \left( {\frac{\sin \beta }{{\cos \theta \tan^{2} \beta }}} \right) - H_{Ro}^{2} \left( {\frac{3r \sin \beta }{\tan \beta } + \frac{3 \sin \beta }{{\cos \theta \tan^{2} \beta }} - \frac{3}{\cos \theta \tan \beta }} \right) + H_{Ro} \left( {\frac{6 r \sin \beta }{\tan \beta } + 3r^{2} \cos \theta \sin \beta - 6 r} \right) - 6 r^{2} \cos \theta \sin \beta . $$
(38)
This last formula calculates the critical height HRo that fulfills the equality between S and V in a right regular truncated pyramid when its three variables Ar (i.e., r), n (i.e., θ) and β are known. It is a cubic equation that can be solved manually or utilizing one of the free software programs available on the Internet (see for example: http://www.1728.org/cubic.htm). As a cubic equation, its four constants a, b, c and d can be calculated as:
$$ a = \frac{\sin \beta }{{\cos \theta \tan^{2} \beta }} $$
(39)
$$ b = - \left( {\frac{3r \sin \beta }{\tan \beta } + \frac{3 \sin \beta }{{\cos \theta \;\tan^{2} \beta }} - \frac{3}{\cos \theta \tan \beta }} \right) $$
(40)
$$ c = \frac{6 r \sin \beta }{\tan \beta } + 3r^{2} \cos \theta \sin \beta - 6 r $$
(41)
$$ d = - 6 r^{2} \cos \theta \sin \beta . $$
(42)
Given the value of HRo, rU can be calculated by applying Eq. 24, consequently the other parameters of such pyramid can be calculated. In this context, it is important to mention that this equality will never exist if:
$$ \frac{{H_{R} }}{\cos \theta \;\tan \beta } > r. $$
(43)

Discussion

All is number, said Pythagoras (580–500 BC) more than twenty centuries ago. Thus, everything finally returns to mathematics. Numbers always give a better understanding for the behavior of a phenomenon as it appears. They also provide a scientific basis for predicting the ways in which things behave. The findings of this work have important applications in different disciplines of advanced building analysis, especially room acoustics as will be discussed in the following section. In summary, these findings:
  • point out special rooms that have a distinct characteristic in acoustics;

  • clarify why rooms that have the same volume and floor area but different shapes have different reverberation times;

  • establish a simple mathematical approach that can help both architects and acousticians to decide early the appropriate room dimensions. These dimensions satisfy the acoustic requirements;

  • help architects and acousticians to answer two important questions:
    • how does S changes in θ (and/or β) and consequently how will the acoustic environment within a room be affected?

    • When deciding upon the appropriate room dimensions that have a given V, is it better to decrease Ar and increase HR or inversely, to increase Ar and decrease HR?

The reverberation time T (the persistence of sound in a particular space after the original sound is produced) is a main indicator in room acoustics. In its simplest form, T can be calculated as:
$$ T = \frac{0.161 \times V}{{\mathop \sum \nolimits_{i = 1}^{n} S_{i} \times \alpha_{i} }}(s) $$
(44)
(Sabine 1993; Cox and D’Antonio 2009), where α is a physical quantity that expresses the ability of a material to absorb the energy of sound.

Upon designing a room acoustically, its T must lay within the permissible limits that depend on the acoustic function of this room. A short T (around 1 s) is recommended especially in speech rooms. As can be concluded from Eq. 44, for a given V, T is a decreasing function of S. Thus, it is recommended to increase room total surface area so as to decrease T.

Based on the findings of this work and according to the shape, remark (1) determines the conditions under which S will take its minimum value among the different rooms that have the same floor area and volume but different θ (and/or β). For example, in rectangular right prisms (or rooms), a room that has θ = 45° (square plan) has the minimum total surface area, thus the maximum T among the other rectangular rooms that have the same volume assuming that α is constant (Elkhateeb 2012).

Following the same rules, remark (2) determines the condition under which S will be minimum among the different rooms that have the same θ (and/or β) and V but different Ar. Such a room also possesses the maximum T and should be avoided (Elkhateeb 2012). Together, remarks (1) and (2) establish a clear methodology that can be applied to select the optimum room dimensions from an acoustic point of view.

In any room, walls are the typical place to install the different acoustic treatments such as absorbing and reflecting materials. Acoustically, in some applications, it is preferable to increase walls ratio so as to insure a good acoustic performance. Utilizing remark (3), RW can be checked easily during the analysis and design phase.

The shape factor Shf is a mathematical indicator that has a direct effect on room acoustics (Elkhateeb 2012). This indicator can be calculated from:
$$ Sh_{f} = \frac{Per}{Ar} \times Dis_{f} , $$
(45)
where Disf is the distortion factor that is the ratio between the total surface area S1 of the examined room and the total surface area S of the reference room RR (i.e., Disf = S1/S). The reference room is a right prismatic room that has the identical base, floor area and volume of the examined room (Fig. 11). The indicator Shf has been suggested as a simple tool that can be used to compare architecturally, on acoustic bases, between rooms that have the same floor area but different shapes. It is preferable to choose the room that has the maximum Shf. When the perimeter of a given shape equals its area according to remark (4), accordingly and under the assumed conditions, Shf = 1. In this case, Shf can’t be used as a comparison tool, and more advanced acoustic analysis will be essential.
Fig. 11

The concept of reference room (RR), Disf = S1/S. aleft Examined room (S1); bright RR (S)

The mean free path l (the average distance travelled by sound ray between successive collisions) is another important acoustic indicator. Under certain conditions l can be calculated from:
$$ l = \frac{4V}{S}(m) $$
(46)
(Kuttruff 2009). According to remark (4), when S equals V, thus, l will be equal to 4 m. In this specific case, T will be equal to (0.161/α) if α is constant for all boundaries of the room. In this last case, if α = 1 (i.e., a perfect absorber), T will be equal to 0.161 s, regardless of the values of both S and V, as long as they are equal.

Conclusions

Applying the same methodology, assumptions and rules that were introduced in the previous Parts I–III of this work, this final Part IV examines the case of the right regular pyramid either complete or truncated. In complete pyramids, the first remark examines the effects of θ and β on S. In the second remark, the minimum total surface area SMin was calculated in two cases, case of variable β and constant θ, and case of constant β and variable θ. In the first case, the critical ratio ωo that corresponds to the critical dihedral angle βo (70.528779°) was calculated. Results showed that ωo depends entirely on θ. The values of ωo were calculated and presented for the common right regular pyramids according to their bases. In the second case, where θ is variable, results showed that SMin occurs when θ → 0 (i.e., cones). In the third remark, the ratio RW was calculated. Results showed that RW is an increasing function of β. Results also showed that RWo is constant (=0.75) regardless the value of (n). In the last remark, the critical room height HRo that fulfills the numerical equality between S and V was calculated. Results indicated the limit under which this equality will never exist.

In a truncated pyramid, the first remark investigates the effects of three independent variable θ, β and HR on S. When θ is variable (whereas HR and β are constants), results showed that S is an increasing function of θ, thus SMin occurs when θ → 0 (i.e., a truncated cone). When β is variable (whereas HR and θ are constants), results showed that S is a decreasing function of β, thus SMin occurs when β → 90 (i.e., a prism). When HR is variable (while θ and β are constants), results showed that S is a decreasing function of HR, thus SMin occurs when HR is maximum. In the second remark, the calculation of RW indicated that it depends on the three variables r, rU and β regardless the values of θ and V. In the last remark, the critical room height HRo that fulfills the numerical equality between S and V was calculated. In this remark also, the limit under which this equality will never exist was presented. Finally, the importance of the findings of this entire work (Parts I–IV) in room acoustics as a branch of the advanced building analysis was presented.

Acknowledgments

The author would like to express his gratitude and sincere appreciation to Editor-in-Chief Kim Williams for her valuable advice and support. Thanks also go to Arch Zinub Najeeb for her continuous help during the preparation of this work.

Copyright information

© Kim Williams Books, Turin 2014

Authors and Affiliations

  1. 1.Department of Architecture, Faculty of Environmental DesignsKing Abdulaziz UniversityJeddahSaudi Arabia
  2. 2.Physics Department, Faculty of ScienceAin Shams UniversityCairoEgypt