# Remarks on the Surface Area and Equality Conditions in Regular Forms Part III: Multi-sided Prisms

## Abstract

Applying the methodology and rules that were previously established in Part I of this work, this part presents the remarks on the mathematical analysis for the regular multi-sided right prisms. According to the shape of their bases, these include shapes from pentagon to circle. The first remark examines the effect of θ on S. The second remark calculates the minimum total surface area (S_{Min}) in two cases, the case of constant θ and the case of variable θ. The third remark calculates walls ratio R_{W} and the critical walls ratio R_{Wo}. The last remark studies the required conditions for the numerical equality in two cases, the case of Per = Ar, and the case of S = V. Finally, the findings of the first group (right regular prisms) are generalized and discussed.

### Keywords

Trigonometry Algebra Differential equations Volume Area Total surface area Perimeter Regular polygons Multi-sided prisms Minimum total surface area Walls ratio Numerical equality## Introduction

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?

Following the same methodology and assumptions that were previously applied, this part investigates the case of the regular multi-sided right prisms.

### Notations

- Ar
: Room floor area (m

^{2})- h
- : The altitude of the triangle, see Fig. 2 (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)

- n
: Number of sides

- 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
: Radius (m)

**r**_{o}: The critical radius, the radius that fulfills (Per–Ar) equality (m)

- 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.

### Room Assumptions

To be contained in a circle.

Its central angle ψ is constant and equal to 360/n.

Through this part, it is assumed that the angle θ, Ar and V are the independent variables whereas Per and S are the dependent ones. Figure 2 shows the terms: θ, ψ, h, L and r in the regular multi-sided shapes.

## The Mathematical Relationships of Multi-sided Shapes

_{Min}and calculate the equality conditions. From the first principles, it can be proved that:

_{R}. In this case, its volume V = (H

_{R}× Ar) can be calculated from:

## Remark 1: Effect of θ on S

## Remark 2: the Minimum Total Surface Area, S_{Min}

Case of constant θ, where both Ar and H

_{R}are variables. Or;Case of variable θ, where both Ar and H

_{R}are constants.

### Case I, Constant θ, Variable Ar and H_{R}

_{Min}occurs when the first derivative of Eq. 14 equals zero, i.e.

**ω**_{o}(see Sect. “Notations”) can be calculated from Eq.16 as:

**ω**_{o}is a function of θ (consequently n). Thus, for every n (i.e. for every regular multi-sided right prism) there is

**ω**_{o}that fulfills S

_{Min}. Eq. 17 also tells that

**ω**_{o}is a decreasing function of θ, therefore, it is an increasing function of n, see Fig. 4. When n reaches

**∞**, then θ → 0°, consequently,

**ω**_{o}= 2, this is the case of a circle. To determine room dimensions that fulfill S

_{Min}in such prisms:

It is worth mentioning in this context that Eq. 17 also applies in the two special cases of equilateral triangle (θ = 60°) and square (θ = 45°). Both shapes can be considered regular multi-sided shapes according to the above definition of this term (see Sect. “Room Assumptions”). If r in Eq. 16 was replaced by its equivalent value of h (**r****=** **2/3*** h* in equilateral triangle, and

**r****=**

*in square), then Eq. 17 will yield*

**h/2**

**ω**_{o}according to Eqs. 14 or 15 (Part I) and Eq. 14 (Part II) (Elkhateeb and Elkhateeb 2014).

_{R}–S) and (Ar–S) depend on

**ω**_{o}which divides these functions into two zones (see Figs. 5 and 6):

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 Fig. 6. 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 the increase in H_{R}will increase S.

### Case II, Variable θ, Constant Ar and H_{R}

_{R}are constants in prisms under consideration. In this case, S reaches its minimum value when the first derivative of Eq. 6 equals zero after replacing both n and r by their equivalent values according to Eqs. 1 and 4, respectively, thus:

Eq. 19 fulfills only if θ equals 0. This result completely agrees with the mathematical axiom that among the different regular shapes that have the same area, the circle (n = **∞**) possesses the minimum perimeter. As a result, among the different regular multi-sided right prisms, a cylinder has the minimum total surface area. Another proof to this result is that Per is an increasing function of θ according to Eq. 6 (consequently, a decreasing function of n according to Eq. 1), thus Per is a decreasing function of n; accordingly S is a decreasing function of n (see Figs. 3 and 9).

## Remark 3: Walls Ratio R_{W}

_{W}can be mathematically defined as:

_{W}and θ resembles the relationship between S and θ (see Fig. 3), thus it is an increasing function of θ, consequently, a decreasing function of n. R

_{W}reaches its minimum value when θ → 0° (circular shapes). To calculate

**R**_{Wo}, the conditions for

**ω**_{o}must be applied, thus, Eq. 21 can be rewritten as:

Consequently, the critical walls ratio **R**_{Wo} (see Sect. “Notations”) in regular multi-sided right prisms is also constant for any θ and equals 2/3.

## Remark 4: Case of Equality

This section calculates two cases of numerical equality in regular multi-sided prisms. The first considers the numerical equality between Per and Ar. The last considers the numerical equality between S and V.

### Case I, Equality of Per and Ar

**r**_{o}(see Sect. “Notations”) can be calculated from Eq. 24 as:

In the special case where n → ∞ (i.e. circular shape), θ → 0. As **cos*** 0* = 1, thus,

**r**_{o}= 2 m.

**ω**_{o}, the numerical equality between Per and Ar also depends on θ according to Eq. 25. The values of

**r**_{o}were plotted in Fig. 7. As can be concluded from this figure, the relationship between θ and

**r**_{o}is similar to the relationship between θ and

*S*(see Fig. 3), where

**r**_{o}is an increasing function of θ (consequently a decreasing function of n).

### Case II, Equality of S and V

**cos***= 1, thus, Eq. 29 will be:*

**0**

**H**_{Ro}tends to

**∞**, i.e., when Ar equals Per according to Eq. 25. Figure 8 represents the relationship between Ar and

**H**_{Ro}calculated from Eq. 27 for a regular pentagonal right prism (θ = 36°). As can be seen from the figure, in the acceptable range,

**H**_{Ro}is a decreasing function of Ar. In this range, the function can be divided into two main zones, zone of rapid decay (when Ar tends to be equal to Per) and zone of slow decay (when Ar is far from this equality).

## Generalization of Results, Group of Regular Traditional Forms

A complete set of mathematical functions that relates Per, Ar and S to the angle θ (or θ and β) was derived.

The effect of θ (or θ and β) on S was investigated.

The minimum total surface area S

_{Min}of the prism and walls ratio R_{W}were calculated.The conditions to fulfill two cases of numerical equality (Per–Ar) and (S–V) were calculated.

When θ is variable, in triangular and rectangular rooms that have the same area, S is a decreasing function of θ until a specific θ where this relationship reverses and S becomes an increasing function of θ. This specific θ = 60° in triangular shapes (i.e. equilateral triangle) and 45° in rectangular shapes (i.e. square). The same fact also applies in trapezoidal shapes but the angle at which the function reverses its direction depends on both θ and β. In regular multi-sided rooms, S is an increasing function of θ (accordingly a decreasing function of n).

*) becomes limited at the higher values of n (n ≥ 10) as can be seen in Fig. 9.*

**ds/dn**When θ is constant (so, Ar is variable) and in prisms under consideration, the mathematical analysis proves that there is a critical ratio **ω**_{o} that makes the total surface area of a room reaches its minimum value. This **ω**_{o} depends on the shape of the base and is a function of θ (or θ and β in trapezoidal shapes). When * ω* <

**ω**_{o}, S becomes a decreasing function of H

_{R}and an increasing function of Ar. This means that any increase in room height will decrease its total surface area. On the contrary, when

*>*

**ω**

**ω**_{o}, S becomes an increasing function of H

_{R}and a decreasing function of Ar. This means that an increase in H

_{R}will increase S.

When θ is variable, the mathematical analysis indicates that walls ratio R_{W} of the prisms under discussion is also a function of θ (or θ and β in trapezoid). This relationship resembles the relationship (θ–S). Hence, in triangular and quadratic prisms, R_{W} is a decreasing function of θ until a specific θ (θ = 60° in triangle and 45° in rectangle) then the function reverses and R_{W} becomes an increasing function of θ. In regular multi-sided rooms, R_{W} is an increasing function of θ (accordingly a decreasing function of n).

**R**_{Wo}is constant in all prisms under consideration and is equal to 2/3. Figures 10 and 11 summarize the findings of this work (in Parts I, II and III) for both S and R

_{W}.

The numerical equality between Per and Ar fulfills when the critical altitude/diagonal or radius (**h**_{o} or **r**_{o}) fulfills. Similar to **ω**_{o}, **h**_{o} (or **r**_{o}) is a function of θ (or θ and β in trapezoid). In the case of circles, **r**_{o} = 2 m.

The numerical equality between S and V also fulfills when the critical room height **H**_{Ro} fulfills. In the acceptable range of the derived formulae to calculate this equality, **H**_{Ro} is a decreasing function of Ar. In the prisms under consideration, the mathematical analysis indicates that for every θ (or θ and β in trapezoid) there is a minimum Ar under which this equality will never exist. This occurs when **H**_{Ro} tends to **∞**, i.e., when Ar equals Per.

## Conclusions

Following the same methodology, assumptions and rules that were applied previously in Parts I and II, this part examines the cases of the regular multi-sided right prisms. According to the shape of their bases, such prisms include shapes from pentagon to circle. The first remark examines the effect of θ on S. In the second remark, the minimum total surface area S_{Min} for the prisms under discussion was calculated in two cases, the case of constant θ and the case of variable θ. In the first case, the critical ratio **ω**_{o} was calculated. Results showed that **ω**_{o} depends entirely on θ. The values of **ω**_{o} were calculated and presented. In the second case, where θ is variable, results showed that S_{Min} occurs when θ → 0 (i.e. cylindrical rooms). The third remark calculates the ratio R_{W}, results showed that R_{W} reaches its minimum value in circular rooms (n = ∞). Results also showed that the critical walls ratio **R**_{Wo} is constant for any n 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 radius **r**_{o} that fulfills Per–Ar equality was calculated. Results showed that **r**_{o} depends entirely on θ. In the second case, the critical room height **H**_{Ro} that fulfills S–V equality was calculated. Results also indicated that for every θ there is a minimum r under which this equality will never exist; this corresponds to **r**_{o} (i.e. Ar = Per). Finally, the results of the first group (regular right prisms) were generalized and discussed.

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