Re-Evaluating the Value of π and Emerging New Concepts for Measuring Areas and Volumes Based on Translations and Rotations of Straight Lines, Planes, and Circles

Indian Journal of Advanced Mathematics (IJAM) ISSN: 2582-8932 (Online), Volume-5 Issue-2, October 2025 81 Retrieval Number:100.1/ijam.A122506010426 DOI: 10.54105/ijam.A1225.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. Re-Evaluating the Value of π and Emerging New Concepts for Measuring Areas and Volumes Based on Translations and Rotations of Straight Lines, Planes, and Circles Nishant Sahdev, Chinmoy Bhattacharya Abstract: The universal constant π has been assigned a nonconverging value of approximately 3.145..., and this non-repeating, non-terminating decimal extends to millions—even trillions—of digits beyond the decimal point, as calculated by the most advanced computational methods. However, for thousands of years, mathematicians across different civilizations have attempted to determine the value of this fundamental constant in various ways. In geometry, angles are commonly expressed in terms of π radians, and π appears prominently in the formulas used to calculate the areas and volumes of curved geometrical figures such as circles, ellipses, spheres, cones, pyramids, and cylinders. Thus, π has become an integral part of how we measure areas and volumes. Ancient scientists, in their efforts to understand and quantify the dimensions of curved or non-linear topological objects, sensed a recurring spatial constant underlying these forms. Over time, this invariant quantity came to be known as “pi.” The pursuit of its exact value has become a continuous endeavour spanning millennia, with each generation refining its estimation using increasingly sophisticated tools and methods. In this article, it is first emphasised that, although the mathematical parameters of area and volume have traditionally been expressed through formulas involving length, breadth, and height, their more profound physical significance has not been adequately explored. This work attempts to redefine area and volume fundamentally in terms of distance or length alone, offering new mathematical formulations for the areas and volumes of basic geometric shapes such as squares, cubes, circles, spheres, and others. Another critical shortcoming of conventional mathematics lies in its neglect of the reciprocal or inverse space of the universe when evaluating the value of π. In recent research, it has been proposed that π itself represents a circle whose radius corresponds to the smallest conceivable length in the universe. Furthermore, the existence of an inverse π—always in conjugation and equilibrium with π—is posited as essential for maintaining the balance of forces in the universe. Accordingly, this article presents a novel derivation of π by simultaneously considering both the cosmos' direct space and the reciprocal (or inverse) space. Manuscript received on 06 October 2025 | Revised Manuscript received on 10 October 2025 | Manuscript Accepted on 15 October 2025 | Manuscript published on 30 October 2025. *Correspondence Author(s) Nishant Sahdev, Department of Research & Development, Austin Paints & Chemicals Pvt. Ltd., Ambika Mukherjee Road, Belghoria, Kolkata (West Bengal), India. Email ID: [email protected], ORCID ID: 00090007-2249-1006 Chinmoy Bhattacharya*, Department of Research & Development, Austin Paints & Chemicals Pvt. Ltd., Ambika Mukherjee Road, Belghoria, Kolkata (West Bengal), India. Email ID: [email protected], ORCID ID: 0000-0002-1962-0758 © The Authors. Published by Lattice Science Publication (LSP). This is an open-access article under the CC-BY-NC-ND license https://creativecommons.org/licenses/by-nc-nd/4.0/ Keywords: Pi (π), Reciprocal Geometry, Area and Volume Redefinition, Geometric Constants, Cosmological Geometry, nonConverging, Planck Length I. INTRODUCTION At the back of the minds of the researchers behind this article, the long-standing misconceptions, illogical philosophies, and the overlooked areas of classical mathematics and geometry were always present — and they served as the driving force behind this research work. i. The areas of different geometric figures are calculated as (length)², which is referred to as the ‘unit square concept’ of area. However, there is no justified or rational mathematical logic or proof supporting this concept. ii. The volumes of different geometric figures are calculated as (length)³, which is referred to as the ‘unit cube concept’ of volume. However, there is no justified or rational mathematical logic or proof supporting this concept. iii. The neglect of the reciprocal or inverse space of the universe in evaluating the value of π has been a significant oversight. In recent research, it has been proposed that π itself represents a circle whose radius corresponds to the smallest conceivable length in the universe. Furthermore, the existence of an inverse π — always in conjugation and equilibrium with π — is posited as essential for maintaining the balance of forces in the universe. Accordingly, this article presents a novel derivation of π by simultaneously considering both the direct space and the reciprocal (or inverse) space of the cosmos, finding its value to be exactly 3. iv. The value of π has conventionally been obtained through practical experiments by dividing the circumference of a circle by its diameter. The fundamental mistake lies precisely here: the rotational or curved length of the circumference has been measured using a linear measuring scale, which is incorrect. The scales of measurement for curved (or rotational) length and linear length must be different. A curved length is a compressed or “squeezed” one, and when it is unfolded or straightened, it becomes a linear length. Across the world, people have continued making the same mistake for thousands of years— measuring the curved length of a circle’s Re-Evaluating the Value of π and Emerging New Concepts for Measuring Areas and Volumes Based on Translations and Rotations of Straight Lines, Planes, and Circles 82 Retrieval Number:100.1/ijam.A122506010426 DOI: 10.54105/ijam.A1225.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. circumference using a linear scale. As established in this research article, for every 1° arc length on the circumference of a circle of unit radius (circumference = 2π), the linear length is greater than the curved length by 0.0008. Half of the circumference of such a circle is π. Now, for a 180° angle, the difference in the circumference’s length comes out to be 0.145 (0.0008 × 180), considering the value of π to be precisely 3 as established in this article, and 3.145 according to the conventional linear measurement, based on the value of π. This difference of 0.145 corresponds to the difference between the traditional value of π (3.145…) and the re-evaluated value proposed in this research article, which is precisely 3. v. Circles, spheres, ellipsoids, and similar shapes belong to the class of geometries known as ‘closed-loop geometries.’ However, their mathematical formulas contain the parameter π, which is non-converging, and as a result, the areas and volumes of these figures also become nonconverging. This is not acceptable, since closed-loop geometries cannot have non-converging areas or volumes. vi. There does exist a ‘smallest length’ in the universe, yet distances are still often expressed in the form of decimals. If the concept of a ‘smallest possible length’ is genuine, then any distance would be an integral multiple of this fundamental length. vii. The topology of the ‘multiplicative inverse’ of circles or spheres is a missing area in geometry and mathematics, and it has never been explored before this research work. A. The value of π is non-converging, with an approximate magnitude of π ≈ 3.145. The 2π radian is equal to a 360degree angle, and the general representation of an angle θ is [1]. θ (𝑖𝑛 𝑟𝑎𝑑𝑖𝑎𝑛)=[π x θ (in degree)] 360 … (1) So, if θ is, for example, 30 degrees, in radian scale it would be (π/12) radian. B. If the arc length of a circle is s and it subtends an angle θ at the centre of the circle, with radius r, [1]. s = rθ … (2) [Fig.1a: The Arc Lengths of a Circle of Radius r as a Function of Angle θ] C. The area (A) of a square of length x is, [1]. A =𝑥2 … (3) [Fig.1b: The Area of a Square of Each Side Length x] D. The area (A) of a rectangle of length = x and breadth = y is, [1]. 𝐴 = (𝑙𝑒𝑛𝑔𝑡ℎ 𝑥 𝑏𝑟𝑒𝑎𝑑𝑡ℎ)= [𝑥𝑦] … (4) E. The volume (V) of a cube of length x is, [1]. 𝑉 =𝑥3 … (5) [Fig.1c: The Volume of a Cube of Length x] F. The volume (V) of a rectangular cube of length, breadth and height being x, y and z respectively, is [1]. 𝑉 = 𝑥𝑦𝑧 … (6) G. The circumference length (C) of a circle of radius r is, [1]. 𝐶 = 2𝜋𝑟 … (7) [Fig.1d: The Circumference of a Circle of Radius r] H. The area of a circle (A) of radius r is, [1]. 𝐴 =𝜋𝑟2 … (8) Indian Journal of Advanced Mathematics (IJAM) ISSN: 2582-8932 (Online), Volume-5 Issue-2, October 2025 83 Retrieval Number:100.1/ijam.A122506010426 DOI: 10.54105/ijam.A1225.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. [Fig.1e: The Area of a Circle of Radius r] I. The volume (V) of a cylinder of base radius r and height h is, [1]. 𝑉 =𝜋𝑟2ℎ … (9) [Fig.1f: The Volume of a Cylinder of Radius r and Height h] J. The volume (V) of a cone of base radius r and height h is, [1]. 𝑉 = 1 3[𝜋𝑟2ℎ] … (10) [Fig.1g: The Volume of a Cone of Radius r and Height h] K. The volume (V) of a sphere of radius r is, [1]. V =4 3[πr3] … (11) [Fig.1h: The Volume of a Sphere of Radius r] L. The ratio of the volume of a cone (of radius r height r), a cylinder (of radius r and height r) and a sphere (of radius r) is [2]. Vcone:Vcylinder: Vsphere = 1: 2: 3 … (12) [Fig.1i: Archimedes’ Presentation of the Ratio of the Volumes of a Cone, a Cylinder and a Sphere] M. The square root of (-1) is imaginary and is represented by i = √−1 … (13) The first important point to note from the above guidelines is that, according to guideline (ii), angles expressed in radians attain a non-converging value due to their dependence on π. However, this is problematic, as angles—being finite geometric entities—should not possess a non-converging or infinite nature. Secondly, regarding guideline (iv), while the area of a square is expressed as the square of its length, i.e., (length)2, the underlying rationale for why area is defined this way has not been adequately explained in conventional mathematics. The above guidelines are being tabulated in Table 1a below: Table 1a: Guidelines of Conventional Mensuration and Mathematics S/No. Guideline On Guideline Detail 1. π parameter Non, having a value of 3.145…. 2. Arc Length of circles (s) S = r θ r is the radius of the circle and θ (in radian) = [π x θ (in degree)]/360 3. Square of side length x Perimeter = 4x Area = x2 4. Rectangle of length = x and breadth =y Perimeter = 2(x +y) Area = xy 5. Homogeneous cube of side length x Total Surface area = 6x2 Volume = x3 6. Circle of radius r Circumference = 4πr Area = πr2 7. Cylinder of height h and radius of the base r Total surface area = 2πr (r + h) Volume = πr2h 8. Cone of height h and radius of the base r Volume = (πr2h/3) 9. Sphere of radius r Surface area = 4πr2 Volume = 4πr3 10. Inter-relationship between the volume of a cone, a cylinder and a sphere (radius r and height h) Vcone: Vcylinder: Vsphere = 1:2:3 11. Imaginary number (i) i = √(-1) Re-Evaluating the Value of π and Emerging New Concepts for Measuring Areas and Volumes Based on Translations and Rotations of Straight Lines, Planes, and Circles 84 Retrieval Number:100.1/ijam.A122506010426 DOI: 10.54105/ijam.A1225.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. The Following new Concepts Should be Considered to Understand Why Area is Expressed as the Square of Length, i.e., (Length)2: i) There does not exist any truly fractional distance in the physical universe. For instance, consider an arbitrary distance of 1.23456 kilometres, which appears to be a fraction in the kilometre scale. However, if we scale down the unit of measurement to hectometers, the same distance becomes 12.3456 hectometers—still a fraction. Continuing this process of scaling down, when the unit is reduced to millimetres (noting that 1km = 105 mm), the distance transforms into 1,234,560 millimetres, which is now an integer value. This example illustrates that any length, which appears as a fractional value at one scale, becomes a whole number at a sufficiently smaller scale. Therefore, through continued unit reduction, any fractional distance ultimately resolves into an integer multiple of a fundamental, indivisible unit of length—the smallest measurable length in the universe. Let this smallest possible length be denoted by dx. Then, any arbitrary length x must be an integral multiple of dx, such that: x = ndx [n is a whole number] … (14) Therefore, the idea of a 'fraction' is, in reality, a conceptual convenience rather than a physical truth. Yet, it continues to play a crucial role in mathematics and daily life due to its practical usefulness. ii) In the physical universe, only the operations of addition and subtraction fundamentally exist. This assertion can be substantiated by examining how multiplication and division can be interpreted as repeated addition or subtraction. 7 times of 5, (5 + 5 + 5 + 5 + 5 + 5 +5) = 35 Or 5 times of 7, (7 + 7 + 7 + 7 +7) = 35 Hence, multiplication is nothing but addition. The operation of division can also be fundamentally interpreted as a process of repeated subtraction. For example, when the number 20 is divided by 4, the result is 5. The physical significance of this operation is that one repeatedly subtracts a constant quantity (in this case, 4) from the original number (20) until the remainder reaches zero. 1st step (20 - 4) = 16 2nd step (16 - 4) = 12 3rd step (12 - 4) = 8 4th step (8 - 4) = 4 5th step (4 - 4) = 0 Hence, as shown, 5 is the result of division. iii) For a polynomial of the form below, where a, b, c, and d are constants (known as coefficients) with no physical dimensions: y = ax3 + bx2 + cx + d … (14a) It is worth noting that if the variable x stands for distance or length, then the first, second, third, and last terms on the righthand side of equation (14a) would represent volume, area, length, and a dimensionless quantity, respectively. So, the question arises: how can these terms be added together? The answer is that whether it is area or volume, they are ultimately forms of ‘distance’; otherwise, they could not be added. If the values of a, b, c, and d, are 1, 2, 3, and 4, respectively, then the above polynomial can be expressed as: Total Distance or length,y = 1.(x.x.x)+ 2 (x.x)+ 3 (x) + 4.1 … (14b) In equation (14b), the unit of the last term on the right-hand side represents the smallest possible length. So, dimensionally, the left-hand side and the right-hand side are the same, both representing distance. Why x3, x2, and x all stand for length is explained below. iv) The following discussion pertains to guidelines (iv) to (vii) concerning the calculation of areas and volumes. To understand these, refer to Figures 1, 2, and 3 below. [Fig.1j: Translation of Line Segments AB & AD Along the y & x Direction and Generating Overlapping Distance in a Square Geometry] Indian Journal of Advanced Mathematics (IJAM) ISSN: 2582-8932 (Online), Volume-5 Issue-2, October 2025 85 Retrieval Number:100.1/ijam.A122506010426 DOI: 10.54105/ijam.A1225.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. [Fig.2: Translation of Line Segments AB & AD Generating Overlapping Distance in a Rectangular Geometry] [Fig.3: Geometrical Presentation of a Homogeneous Cube of Length x] In Figure 1i, it is shown that in a square, the line segment AB of length x (along the direction of the x-axis) translates toward the opposite line segment CD through points A and B (along lines AD and BC) x number of times (or its own number of times). Since in each translation it travels a distance x, the total distance travelled is: x, x, x, x, x times for AD x, x, x, x, x times for BC 𝐻𝑒𝑛𝑐𝑒 𝑡𝑜𝑡𝑎𝑙 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑐𝑜𝑣𝑒𝑟𝑒𝑑 = 2(𝑥.𝑥)= 2𝑥2 … (15) Now, there has to be a distinction between length and area. In the case of length, there is no overlapping of distances. However, in the case of an area, there must be some overlapping of distances. What exactly is this overlapping? This concept is illustrated in Figure 3a. The line AD, oriented along the y-axis, travels x times toward the opposite line BC— resulting in an overlap with the area described in equation (15)—such that the total distance travelled by AD would be: x, x, x, x, x times = x2 along AB x,x,x,x…………………x times for CD =2𝑥2 … (16) The area of a square would be the sum of equations (15) and (16), such that: The area of square ABCD = (2𝑥2 + 2𝑥2) = 4𝑥2 … (17) The circumference or perimeter of the square is the total distance around its four sides, which is 4 times the side length. The mathematical formula for the circumference of a square =4x … (17a) In the case of a rectangle, as shown above in Figure 2, with length = x and breadth = y, the line x would translate along the line y, x times. Conversely, the line y would translate along the line x, y times, such that the total distance travelled would be: Line x, y, y, y, y, 2x times = 2 xy Line y, x, x, x, x, 2y times = 2xy So,the area of the said rectangle = (2xy + 2xy) = 4xy … (18) The above derivation of the area of a square regarding translations of the sides of the square along each other could also be derived in the following manner, as shown in Figure 3a below. [Fig.3a: The Consecutive Decreases in the Area of a Square Upon the Reduction of Length and Breadth Each by an Infinitesimal Small Distance dx (Smallest Possible unit Length of the Universe), the Planck Length [4] and the Area of the Square Converging to Zero] As shown in Figure 3a, for example, a person is walking along the perimeter of a square with each side length x. After completing the first round of walking, the person starts the second round along the sides, but the length of each side of the square becomes (x-dx). Here, dx stands for the smallest possible unit length of the universe. In the 3rd round of walking, the person walks through a square Re-Evaluating the Value of π and Emerging New Concepts for Measuring Areas and Volumes Based on Translations and Rotations of Straight Lines, Planes, and Circles 86 Retrieval Number:100.1/ijam.A122506010426 DOI: 10.54105/ijam.A1225.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. whose length of each side is (x-2dx). Suppose the walk continues like this; for example, in the xth round, four rounds off the walk. In that case, the person reaches a central point as shown in Figure 4.3a. and the total distance is measured from there. The area of the square formed by the person walking from the 1st round to the 2nd round to the 3rd round ... to the xth round is calculated. The total distance walked by the person up to the xth round of walk is, Total Distance = S = [ 4x + 4(x−dx)+ 4(x − 2dx)+ 4(x−3dx)+ 4(x− 4dx)+ ……… + 4(x−xdx)] … (18𝑎) Splitting equation …, one gets, S = (4x + 4x + 4x + 4x + ⋯……....x no of terms) − 4dx [{1 + 2 + 3 + …. (x − 1) no.of terms}] = 4𝑥2− 4dx.x−1 2[ 2 + (x−2)] = 4𝑥2− 4dx.x−1 2[ (x)] = 4𝑥2−4dx.𝑥2 2[1,neglected] … (18𝑏) = 4𝑥2−2𝑥2𝑑𝑥 = 4𝑥2[dx containing term neglected] … (18𝑐) [The smallest possible length of the universe (dx) is one Planck length and in meter scale its value is 1.6 x 10-36m and so the value of the second term of equation (18b) for a tremendous value of x (in the order of diameter of the earth, 107meter, the value of the said second term would be, (2 x 1014 x 1.6 x 10-36) meter = (3 x 10-22) meter and hence it could be neglected. However, for a value of x being considerably very low, as for example, x = 0.001 meter, the value of the second term would be about to be (3 x 10-42) while the value of the 1st term of equation (18b) would be (4x 10-6) and which is 1036 times than the first term]. In Figure 3, a homogeneous cube is shown, with each side having length x. From the figure, it can be observed that there are 3 squares along each of the x, y, and z directions, as indicated by the tick marks. Each square has an area of 4x2, and the translations of the square would take place in the following manner: The distance travelled by a square of area 4x2 along the x-axis along a length x =(4𝑥2𝑡𝑖𝑚𝑒 2𝑥) =(2x,ex,3x, 4𝑥2𝑡𝑖𝑚𝑒) = 4𝑥2.2𝑥 = 8𝑥3 … (19) The distance travelled by a square of area 4x2 along the y-axis along a length 2x =(4𝑥2𝑡𝑖𝑚𝑒 2𝑥) =(2x,2x,2x, 4𝑥2𝑡𝑖𝑚𝑒) = 4𝑥2.2𝑥 = 8𝑥3 … (20) The distance travelled by a square of area 4x2 along the z-axis along a length x =(4𝑥2𝑡𝑖𝑚𝑒 2𝑥) =(2x,2x,2x, 4𝑥2𝑡𝑖𝑚𝑒) = 4𝑥2.2𝑥 = 8𝑥3 … (21) So, the volume of the cube would be the summation of the distances of equations (19), (20) and (21) and hence the volume would be, Volume of the cube =(8𝑥3+8𝑥3 +8𝑥3 … (22) The total surface area of the cube = 6 x 4𝑥2 =24𝑥2 … (22𝑎) [To note that each of the 6 surfaces of the cube, there would be two numbers of surfaces, the outer surface and the inner surface. In conventional mensuration only one surface is being considered]. Some people may find the newly proposed mathematical formulas difficult to accept or justify. To address this, the following pointwise discussion aims to clearly demonstrate the extent to which the existing mathematical formulas for areas and volumes may be flawed or incomplete. i) Area and volume are all concepts of overlapping of space along x, y and z directions. ii) The areas and volumes are the summation of overlapped distances only, so areas and volumes are ultimately converged to distance only. In the broadest sense, every physical variable of the universe is ‘distance’ only or a multiple of ‘distance or length’. iii) The mathematical operations are only two: ‘addition or summation’ and ‘subtraction’ in the universe. Any mathematical operation has to be shown either in the form of ‘summation’ or ‘subtraction’. iv) Once people are convinced by the points mentioned above, the justifiability of the conventional mathematical formulas can be examined. The conventional formulas for the perimeter and area of a square with each side of length x are 4x and x2, respectively. Now when x is taken to be 1, the perimeter turns out to be four and area be 1. Since the perimeter and area both are distances only, and area is the summation of the overlapping distances over the x and y directions, how come the area is less than the perimeter of a square? The newly derived formula of the area of a square is 4x2 (instead of x2 of the conventional formula), and the formula of perimeter remains the same (4x). For a value of x = 1, the area becomes 4, and the perimeter is also 4. This is a believable and convincing result. When x = 2, the perimeter would be Indian Journal of Advanced Mathematics (IJAM) ISSN: 2582-8932 (Online), Volume-5 Issue-2, October 2025 87 Retrieval Number:100.1/ijam.A122506010426 DOI: 10.54105/ijam.A1225.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. (4 x 2) = 8 and the area becomes 4 x 22 = 16. In the case of x = 1, the perimeter and area of a square are the same only because 1x1 = 1. For any value of x greater than 1, the area is always greater than the perimeter. As per the conventional formula, when x = 2, the perimeter and area are obtained to be 8 and 4, respectively. So, the perimeter being larger than the area by magnitude is not acceptable. v) For a homogeneous cube of length x, the conventional mathematical formulas of surface area and volume are 6x2 and x3, respectively. When x = 1, the surface area becomes six and the volume becomes 1. According to the newly developed formula, the formulas for surface area and volume are 24x2 and 24x3, respectively. When x = 1, the surface area and volume become both 24. When x = 2, the surface area becomes 96, and the volume becomes 192, according to the newly developed formula. According to the conventional formula, when x = 2, the surface area and volume are 24 and 8, respectively. Therefore, the volume being less than the surface area is not acceptable again. For a triangle as shown in Figure 3b below, the base length B = (B1 + B2). Now, the area of the rectangle on the left side of the figure is 4B1H (H is the height of the triangle). So, the area of the triangle portion on the left side would be half of the rectangle and would be 2(B1H). Similarly, the area of the triangular section on the right side would be B2H. So, the area of the entire triangle would be, [Fig.3b: Geometrical Presentation of a Typical Scalene Triangle of Base Length B and Height H] Area of the triangle ABC = (2𝐵1H + 2𝐵2H)= 2H (𝐵1+𝐵2) = 2BH = 2(Base x height) … (22𝑏) Hence, the mathematical formula for the area of a triangle would be, A (triangle)= 2 (base x height) … (22𝑐) So, in conclusion, i) A square or a rectangle contains two numbers of square or rectangle (of the conventional geometry) in it, each of which is 4x2 or 4xy for square and rectangle, respectively. So, the actual area becomes 4x2 and 4xy, respectively, for a square and a rectangle. ii) A Cube contains three cubes (of the conventional geometry), and a rectangular cube contains three rectangular cubes in it. The volume would be 8x3 and 8xyz for the regular and rectangular cubes, respectively. So, the total volume would be 24x3 and 24xyz, respectively. iii) A triangle contains two triangles and its area is 2 (base x height). II. NON-LINEAR GEOMETRY In non-linear geometry, the universal parameter π plays a central role—for example, in computing the circumference of circles, the areas of circles, and the volumes of cylinders and spheres, etc. In Figure 4 below, the derivation of the circumference of a circle is shown. [Fig.4: Rotation of a Line Segment r Through 2π Radians, Resulting in the Evolution of the Circumference of a Circle] In Figure 4, a circle of radius r is shown. Since the circumference is a length, the concept of overlapping distances—previously discussed in the context of area and volume—does not apply here. As shown in Figure 4, when the radius r completes one full rotation along the circumference, it travels through an angular distance of 2π radians. Now, if the radius r rotates r times along the circumference, then the total distance travelled would be: 2π,2π,2π,2π r times and hence the total length travelled for this entire journey would be (2π x r) = 2πr and hence, The circumference of a circle of radius r = 2πr … (23) In Figure 5, the radii of a circle (r) are shown along both directions in 2D space. [Fig.5: Rotation of the Radius r of a Circle Along the x and y Directions by 2π Radians] When the radius r rotates along the x-axis r times through the circumference of length 2πr, the distance travelled, 2πr,2πr,2πr,r times Hence the total distance travelled would be,(r x 2πr) =2π𝑟2 … (24) For the rotation of the radius r along the y direction, another area 2πr2 will be added to equation (24). So, the mathematical formula of the area of a circle would be = 2 x 2π 𝑟2= 4π𝑟2 … (25) For a sphere of radius r, a circle of area 4πr2, if it makes a single end-to-end rotation in a 3D plane (through the circumference of the circle 2πr), as shown in Figure 6, the Re-Evaluating the Value of π and Emerging New Concepts for Measuring Areas and Volumes Based on Translations and Rotations of Straight Lines, Planes, and Circles 88 Retrieval Number:100.1/ijam.A122506010426 DOI: 10.54105/ijam.A1225.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. distance travelled would be 2πr. Now, if the said end-to-end rotation does take place 4πr2 times. [Fig.6: Formation of a 3D Sphere Arising Out of the Rotation of 3 Circles (xy, yz & xz planes) through a Circular Plane of Length 2πr] Distance travelled = (4πr2) x (2πr) = 8π2 r3 So, the rotation of the radius r along the x, y and z directions has to be considered. So, the volume would be 3 times 8π2 r3. Hence, the mathematical formula of a Sphere of radius r would be = 24π2 r3 … (26) [It is important to note that a two-dimensional plane must possess a thickness equal to the smallest possible length in the universe; otherwise, the plane would not physically exist. Under such a condition, both an outer surface area and an inner surface area must co-exist.] For a cylinder of base radius 2r and height h, as shown in Figure 6a, there does exist a rectangle of length 2r and breadth h. So, the area of the rectangle would be (4 x 2r x h) = 8rh. This rectangle rotates on its axes once; the distance travelled would be 2πr. Now, when the said rectangle would rotate 4π times, the total length travelled would be (2πr x 8rh) = 16πr2h. One needs to consider the rectangles in the cylinder (along x-y, x-z & y-z respectively), and each will contribute 16πr2 h, and hence it will be (3x16πr2 h) = 48π r2h. [Fig.6a: Topological Presentation of the Formation of a Cylinder Upon the Rotation of a Rectangular Plane BEFC by 360-Degree Rotation on Axes AD] So, the mathematical formula of the volume of a cylinder would be, V = 48π r2h … (27) A cylinder has two circular bases, each contributing 4πr2 to its surface area. The curved surface of the cylinder can be considered as a rectangle with length 2πr2 and width h. So, its area would be: The contribution to the base circles to the surface area = 2 x 4πr2 = 8πr2 … (27𝑎) The contribution of the curved surface = 8πhr … (27b) So, the mathematical formula of the total surface area of a cylinder would be =8π𝑟2+8π𝑟ℎ =8π𝑟(𝑟+ℎ) … (27𝑐) The volume of a sphere of radius r is composed of two hemispheres, each of height r, as shown in Figure 6b. If one of these hemispheres is uniformly and homogeneously compressed from the top and merged with the ground level, it will form a flat circular shape. The radius of this new circle becomes r+r/4, since the height rrr gets equally distributed among the four radii, as shown in Figure 6c. Therefore, the radius of the newly formed circle is (5r/4). This transformation is illustrated in Figures 6b and 6c. The area of the circle with radius (5r/4), according to the newly derived formula in this article, would be: [Fig.6 (b, c & d): Topological Presentation of the Formation of a Circle Upon Merging a Hemisphere to the Ground by Pressing from the Top as Being Shown] Surface area of one hemisphere = area of the new circle =4π (5𝑟 4)2= 6.25π𝑟2 So the total surface area of the entire sphere would be =2x6.5π𝑟2= 12.5π𝑟2 … (27𝑑),𝑠𝑜 the mathematical formula of the surface area of a sphere is = 12.5π𝑟2 … (27𝑒) A cone, as shown in Figure 7, contains a central triangle with base radius r, height h, and slant length L. The surface area of the cone would be the sum of the area of the base circle and the curved surface area. The circumference (or length) of the base circle is 2πr, which Indian Journal of Advanced Mathematics (IJAM) ISSN: 2582-8932 (Online), Volume-5 Issue-2, October 2025 89 Retrieval Number:100.1/ijam.A122506010426 DOI: 10.54105/ijam.A1225.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. consists of 2πr segments, each corresponding to the smallest possible length in the universe. Each such smallest segment forms a triangle whose height is L. Therefore, the total area of the entire curved surface of the cone is the summation of an infinite number of such smallest triangle areas. [Fig.7: Formation of a Cone Upon Rotation of the Base of a Triangle ABC by 2π Radians (BC = r, AB = h)] Now the area of each of such smallest-to-smallest triangles would be, as per the newly developed formula of a triangle, which is, 2(base x height) = 2(1 x L) = 2L [where 1 is the smallest possible unit base length and L is the height and for such a small length, the linear length and the curved length would virtually be the same). So, the curved surface area would be 2πr times 2L, So,the curved surface area = (2πr x 2L) = 4πrL … (28) Now the area of the base circle,as per new formula = 4π𝑟2 … (29) So, the total surface area of the cone would be, = 4π𝑟2+4πrL 𝑂𝑟,= 4πr (r+L) … (30) Now, the volume of the cone, as shown in Figure 7, can be understood by considering a triangle with a base length of 2r and height h, which rotates through a circular path of length 2πr by 360 degrees. The area of the triangle = 2 (2r x h)= 4rh … (31) So, the triangle would rotate 4rh times through the circular path of length 2πr, and one has to consider three such rotating triangles along the x–y, x–z, and y–z planes.The volume of the cone would be = 3 x 4rh x 2πr = 24π𝑟2ℎ … (32) The following Table 1b shows the derived new formulas for the area and volumes of various geometrical figures compared to the conventional formulas. Table 1b: The Newly Derived Formulas of the Geometrical Figures Vis-à-Vis the Conventional Formulas Geometrical Figure Conventional Mathematical Formula of Area / Surface Area / Perimeter / Circumference Newly Developed Mathematical Formula of Area / Surface Area / Perimeter / Circumference Conventional Mathematical Formula of Volume Newly Developed Mathematical Formula for Volume Square of each side of length x Perimeter = 4x Perimeter = 4x (retained the conventional formula) - - Area = x² Area = 4x² Rectangle of sides x and y Perimeter = 2(x + y) Perimeter = 2(x + y) (retained back the conventional formula) - - Area = xy Area = 4xy Triangle of base length B and height h Area = ½ (B × h) Area = 2 (B × h) - - Homogeneous Cube of length x Surface Area = 6x² Surface Area = 24x² Volume = x³ Volume = 24x³ Rectangular Cube of side length, breadth and height x, y & z, respectively Surface Area = 6(xy + xz + yz) Surface Area = 24(xy + xz + yz) Volume = xyz Volume = 24xyz Circle of radius r Circumference = 2πr Circumference = 2πr (retained) - - Area = πr² Area = 4πr² Cylinder of radius r and height h Surface Area = 2πr(r + h) Surface Area = 8πr(r + h) Volume = πr²h Volume = 48πr²h Sphere of radius r Surface Area = 4πr² Surface Area = 12.5πr² Volume = (4/3)πr³ Volume = 24π²r³ Cone of radius r, height h & slant length L Surface Area = πr² + πrL = πr(r + L) Surface Area = 4πr² + 4πrL = 4πr(r + L) Volume = (1/3)πr²h Volume = 24(πr²h) In the end, the question arises: how can a rotation occur π times, an integral multiple of π times, or nπr times—especially when π itself is a non-converging value? The next section of this article will address and answer this question. III. DETERMINATION OF THE VALUE OF Π The area of a square with each side length being √2r and the length of the diagonal being r forms an outer circle with radius r as shown in Figure 7a & 7b below. In fact, half of the diagonal segment length is the length of OA as it rotates by 360 degrees in the plane of the square to form the outer circle. The higher the area of the square (or the higher the value of r), the greater the area of the outer circle, as shown in Figures 7a & 7b.