Inner Perimeter: inner_perimeter = n_sides * r * sin(π / n_sides) Outer Perimeter: outer_perimeter = n_sides * r * tan(π / n_sides) Estimate of π: π_estimate = (inner_perimeter + outer_perimeter) / (2 * r) Value of N: 107706692 Estimated value of π: 3.141592653589793
The formulas provided are used to estimate the value of the mathematical constant π (pi). This method is based on approximating a circle with a polygon.
Here, "n_sides" represents the number of sides of the polygon, "r" is the radius of the circle, and "sin" and "tan" denote the sine and tangent functions, respectively. In this context, π refers to the value of pi. Let's break down these formulas step by step to calculate the estimate of π. Given: N = 107706692 (assumed to be the number of sides of the polygon, i.e., n_sides) r = 1 (for simplicity, we can set the radius of the circle, r, to 1) According to the formulas, we have: Inner Perimeter: inner_perimeter = n_sides * r * sin(π / n_sides) Outer Perimeter: outer_perimeter = n_sides * r * tan(π / n_sides) Substituting n_sides = 107706692 and r = 1, we get: inner_perimeter = 107706692 * sin(π / 107706692) outer_perimeter = 107706692 * tan(π / 107706692) Since n_sides is very large, π/n_sides will be very close to zero. When the angle approaches zero, both sin(x) and tan(x) can be approximated as x (in radians). Therefore, we can further simplify to: inner_perimeter ≈ 107706692 * (π / 107706692) = π outer_perimeter ≈ 107706692 * (π / 107706692) = π Finally, the estimate of π is: π_estimate = (inner_perimeter + outer_perimeter) / (2 * r) Substituting, we calculate: π_estimate = (π + π) / (2 * 1) π_estimate = (2π) / 2 π_estimate = π This means that using such a large n_sides, both the inner and outer perimeters are very close to the actual circumference of the circle, so the estimate of π is also very close to its true value. In fact, due to the large n_sides, this estimation method can provide a very accurate value of π, almost equal to the actual π ≈ 3.141592653589793. This explains why N = 107706692 is used as the number of sides of the polygon to estimate the value of π, as it ensures a high precision result. Here, there are two formulas for calculating the inner and outer perimeters of a regular polygon, along with a π estimation formula based on these perimeters.
First, there are two key perimeter formulas: Inner Perimeter: inner_perimeter = n_sides * r * sin(π / n_sides) Outer Perimeter: outer_perimeter = n_sides * r * tan(π / n_sides) Where n_sides is the number of sides of the polygon, and r is the distance from the center of the polygon to its vertices (i.e., the radius or circumradius). Next, there is a formula to estimate π: π_estimate = (inner_perimeter + outer_perimeter) / (2 * r) Now, let's substitute these formulas and simplify to verify their correctness. After substitution, we get: π_estimate = (n_sides * r * sin(π / n_sides) + n_sides * r * tan(π / n_sides)) / (2 * r) Since n_sides and r appear in both the numerator and denominator, they can be canceled out (assuming n_sides ≠ 0 and r ≠ 0): π_estimate = (1 / 2) * [sin(π / n_sides) + tan(π / n_sides)] * n_sides Note: The simplification process here does not further reduce the expression because the internal sine and tangent functions are not multiplied or divided by n_sides for simplification. However, it is important to note that r has been canceled, indicating that the estimate does not depend on the specific size of the radius, provided the polygon is similar, meaning the number of sides and the shape ratio are the same. Nevertheless, this expression is not directly equal to π. In fact, it is an estimate of π based on the approximation of the perimeter of a regular polygon. As n_sides increases (i.e., the polygon approaches a circle), this estimate gets closer to the true value of π. To verify this, you can choose a very large value for n_sides and use a calculator to compute this estimate. It will be found that as n_sides increases, the estimated value approaches the true value of π. But it is important to understand that this formula itself is not an exact expression for π; rather, it is an estimation method based on polygon approximations.
This is a detailed summary: Purpose: To estimate π (pi) using the inner and outer perimeters of a polygon until the estimate stabilizes within a specified tolerance range. Main Functionality: calculate_pi(n_sides: usize): Input: n_sides (the number of sides of the polygon, type usize) Output: Returns a tuple containing two f64 values representing the lower bound (pi_lower) and upper bound (pi_upper) of the estimate for π. Operation: Set the radius to 1.0. Use the sine function to approximate the inner perimeter of the polygon (inner_perimeter). Use the tangent function to approximate the outer perimeter of the polygon (outer_perimeter). Return the adjusted inner and outer perimeters (divided by 2.0 * radius) as the lower and upper estimates for π. main(): Initialization: Set the initial number of sides of the polygon (n_sides) to 3. Set the tolerance range (TOLERANCE) to 1e-15. Looping Operation: Call calculate_pi(n_sides) to get the lower and upper estimates for π. Check if the difference between the estimates is less than the specified tolerance range. If it is: Print the current n_sides value and the average π value based on the lower and upper estimates, then exit the loop. If it is not: Increment the n_sides value by 1 and continue to the next iteration. Key Points Summary: Use the perimeters of inner and outer polygons to estimate π. Rely on a loop to continuously increase the number of polygon sides to improve estimate accuracy. Set a tolerance range (TOLERANCE) to determine when to stop the loop and output results. The result includes the minimum number of polygon sides (n_sides) that meets the tolerance range and the corresponding estimate of π.
Objective: To estimate the value of π (pi). Method: Using the perimeters of inner and outer polygons to approximate the circumference of a circle (i.e., 2πr). Mathematical Logical Steps: Inner Perimeter of the Polygon: Concept: When a polygon is inscribed in a circle, each vertex of the polygon lies on the circle. Goal: Use the perimeter of the inscribed polygon to estimate the lower bound of the circle's circumference (2πr). Mathematical Derivation: Assume the radius r = 1.0 (for simplification). The number of sides of the polygon = n_sides. Each internal angle corresponds to a central angle θ = 2π/n_sides (since the sum of all central angles is 2π). Using the half-angle formula, the length of each side of the inscribed polygon is approximately 2r * sin(θ/2) (because θ/2 is the half-angle). Since θ = 2π/n_sides, then θ/2 = π/n_sides. Each side length ≈ 2 * 1.0 * sin(π/n_sides). The inner perimeter (inner_perimeter) ≈ n_sides * 2 * 1.0 * sin(π/n_sides). Implementation: n_sides as f64 * 2.0 * radius * (PI / n_sides as f64).sin(). Outer Perimeter of the Polygon: Concept: When a polygon is circumscribed around a circle, each side of the polygon is tangent to the circle. Goal: Use the perimeter of the circumscribed polygon to estimate the upper bound of the circle's circumference (2πr). Mathematical Derivation: Similarly, assume the radius r = 1.0. The number of sides of the polygon = n_sides. Each external angle corresponds to the same central angle θ = 2π/n_sides. Using the tangent function, the length of each side of the circumscribed polygon is approximately 2r * tan(θ/2) (due to properties of tangent lines). Since θ = 2π/n_sides, then θ/2 = π/n_sides. Each side length ≈ 2 * 1.0 * tan(π/n_sides). The outer perimeter (outer_perimeter) ≈ n_sides * 2 * 1.0 * tan(π/n_sides). Implementation: n_sides as f64 * 2.0 * radius * (PI / n_sides as f64).tan(). Estimating the Value of π: Concept: When n_sides is sufficiently large, the perimeters of the inner and outer polygons will approach the circumference of the circle (2πr). Implementation: Calculate the average of the inner and outer perimeters as the estimate for π. Formula: (pi_lower + pi_upper) / 2.0 (where pi_lower comes from inner_perimeter / (2.0 * radius), and pi_upper comes from outer_perimeter / (2.0 * radius)). Loop Termination Condition: Concept: Stop the loop when the difference between the inner and outer perimeters (i.e., the error in the π estimate) is less than the specified tolerance range (TOLERANCE). Implementation: if (pi_upper - pi_lower).abs() < TOLERANCE {...} Through these mathematical logical steps, we use the perimeters of inner and outer polygons to approximate and estimate the value of π until the specified precision is achieved.
Verification Objectives: Convergence: Confirm that as n_sides increases, the estimated value approaches the true value of π. Accuracy: Verify that the estimated value can reach the true value of π within a given tolerance range (TOLERANCE). Mathematical Analysis and Verification: Inner Perimeter of the Polygon (inner_perimeter): Simplified Function Approximation: When x approaches 0, sin(x) ≈ x (first Taylor series approximation). Applicable in This Case: When n_sides is sufficiently large, π/n_sides approaches 0, so sin(π/n_sides) ≈ π/n_sides. Approximate Inner Perimeter: inner_perimeter ≈ n_sides * 2 * 1.0 * (π/n_sides) = 2π (consistent with the true circumference of the circle). Outer Perimeter of the Polygon (outer_perimeter): Tangent Function Approximation: When x approaches 0, tan(x) ≈ x + x³/3 (first Taylor series approximation, ignoring higher-order terms). Applicable in This Case: When n_sides is sufficiently large, π/n_sides approaches 0, so tan(π/n_sides) ≈ π/n_sides + (π/n_sides)³/3. Approximate Outer Perimeter: outer_perimeter ≈ n_sides * 2 * 1.0 * (π/n_sides + (π/n_sides)³/3) = 2π + (2π³)/(3n_sides²). Observation: As n_sides increases, the additional term (2π³)/(3n_sides²) approaches 0, thus the outer_perimeter also approaches the true circumference of the circle, 2π. Convergence Analysis: Inner Polygon: As n_sides increases, the inner_perimeter approaches the true value of π from below. Outer Polygon: As n_sides increases, the outer_perimeter approaches the true value of π from above. Conclusion: Both converge to the true value of π, and the average estimated value also approaches the true value as n_sides increases. Accuracy Analysis: Upper Bound of Error: Due to the additional term (2π³)/(3n_sides²) in outer_perimeter, the maximum error can be estimated. Given Tolerance Range (TOLERANCE = 1e-15): The lower limit of n_sides can be found by solving (2π³)/(3n_sides²) < 1e-15. Calculation Result: Assuming floating-point precision, n_sides needs to be greater than 14,000 to achieve the specified tolerance range. Verification Results: Convergence: Verified that the estimated values approach the true value of π as n_sides increases. Accuracy: Verified that, under the given tolerance range (TOLERANCE = 1e-15), n_sides must be greater than 14,000 to achieve the required accuracy. Adjustment Suggestions (Optional): Consider increasing the initial value of n_sides to speed up convergence. The TOLERANCE value can be dynamically adjusted to balance accuracy and performance.
Inner Perimeter of the Polygon (inner_perimeter): Visualization: Imagine a regular polygon inscribed in a circle (e.g., an equilateral triangle, square, pentagon, etc.). The radius of the circle is r (set to 1.0). Geometric Observation: Each side of the polygon is tangent to the circle (meaning the endpoints of the sides lie on the circle). Connect the midpoint of each side to the center of the circle, forming a series of radius segments. These radius segments divide the circle into sectors, with each sector having a central angle equal to half of the polygon's internal angle. Formula Derivation: The arc length of each sector (which corresponds to the length of one side of the polygon) is approximately equal to the radius r multiplied by the sine of the central angle (using the small angle approximation since the central angle is small): Arc length ≈ r * sin (central angle) The central angle = π/n_sides (since the sum of all central angles is 2π and the polygon has n_sides). Arc length ≈ r * sin(π/n_sides) The inner perimeter (inner_perimeter) is the sum of all arc lengths: inner_perimeter ≈ n_sides * r * sin(π/n_sides). Outer Perimeter of the Polygon (outer_perimeter): Visualization: Imagine a regular polygon circumscribed around a circle (e.g., an equilateral triangle, square, pentagon, etc.). The radius of the circle is r (set to 1.0). Geometric Observation: Each vertex of the polygon lies on the circle. Connect each vertex to the center of the circle, forming a series of radius segments. These radius segments divide the circle into sectors, with each sector having a central angle equal to half of the polygon's external angle. Formula Derivation: The arc length of each sector (which corresponds to the length of one side of the polygon) is approximately equal to the radius r multiplied by the tangent of the central angle (using the small angle approximation since the central angle is small): Arc length ≈ r * tan (central angle) The central angle = π/n_sides (since the sum of all central angles is 2π and the polygon has n_sides). Arc length ≈ r * tan(π/n_sides) The outer perimeter (outer_perimeter) is the sum of all arc lengths: outer_perimeter ≈ n_sides * r * tan(π/n_sides). Geometric Understanding: As n_sides increases, the inner perimeter (inner_perimeter) approaches the circumference of the circle (2πr) because the sides of the polygon get closer to the arc of the circle. Similarly, as n_sides increases, the outer perimeter (outer_perimeter) also approaches the circumference of the circle (2πr) because the vertices of the polygon get closer to the arc of the circle. This formula utilizes these two approximations by averaging the inner and outer perimeters of the polygon to estimate the value of π.
Inner Perimeter of the Polygon (inner_perimeter): Perspective: Imagine a circle (radius r) divided into n_sides equal sectors. Each sector has a central angle of dθ = 2π/n_sides. Each side of the inscribed polygon can be viewed as a chord connecting two adjacent sector vertices. Chord Length as an Integral Expression: The chord length (ds) can be seen as an infinitesimal displacement along the circumference of the circle. Using polar coordinates, the differential element of chord length (ds) can be written as: ds = r * dθ. Substituting dθ with 2π/n_sides gives: ds = r * (2π/n_sides) * dθ' (where dθ' is an infinitesimal angle change proportional to dθ). Integral of the Inner Perimeter: The inner perimeter (inner_perimeter) can be considered as the total length of all chords, represented as an integral: inner_perimeter = ∫[0, 2π] r * dθ. Dividing dθ into n_sides equal parts results in: inner_perimeter ≈ ∑[i=1, n_sides] r * (2π/n_sides) * sin(π/n_sides) (using the small angle approximation, sin(dθ) ≈ dθ). Formula: n_sides * r * sin(π/n_sides (corresponding to the discrete version of the integral). Outer Perimeter of the Polygon (outer_perimeter): Perspective: Similarly, imagine a circle (radius r) divided into n_sides equal sectors. Each sector has a central angle of dθ = 2π/n_sides. Each side of the circumscribed polygon can be viewed as a tangent connecting two adjacent sector vertices. Tangent Length as an Integral Expression: The tangent length (dt) can be seen as an infinitesimal displacement along the circumference. Unlike chord length, tangent length uses the tangent function: dt = r * tan(dθ). Substituting dθ with 2π/n_sides gives: dt = r * tan(2π/n_sides) * dθ' (where dθ' is an infinitesimal angle change proportional to dθ). Integral of the Outer Perimeter: The outer perimeter (outer_perimeter) can be considered as the total length of all tangents, represented as an integral: outer_perimeter = ∫[0, 2π] r * tan(dθ). Dividing dθ into n_sides equal parts yields: outer_perimeter ≈ ∑[i=1, n_sides] r * tan(π/n_sides) (using the small angle approximation, tan(dθ) ≈ dθ + (dθ)³/3, but higher-order terms are omitted here). Formula: n_sides * r * tan(π/n_sides (corresponding to the discrete version of the integral). Understanding the Perspective: The integral for the inner perimeter can be viewed as the total chord length along the circle, approximated using the sine function. The integral for the outer perimeter can be viewed as the total tangent length along the circle, approximated using the tangent function. This formula represents the discrete version of these integrals, using n_sides equal sectors to approximate the circumference of the circle.
Formulas: inner_perimeter = n_sides * r * sin(π/n_sides) outer_perimeter = n_sides * r * tan(π/n_sides) π_estimate = (inner_perimeter + outer_perimeter) / (2 * r) Time Complexity Analysis: Sin and Tan Functions: The time complexity of these functions is generally considered O(1), as their computation time does not depend on the input size (n_sides). However, note that in some implementations, these functions may involve calculations of polynomials or rational functions, which could lead to higher time complexities (e.g., O(log n) or O(n)). In this analysis, we assume efficient O(1) implementations are used. Multiplication and Division Operations: The time complexity for these operations is also O(1), as their computation time is independent of the input size (n_sides). Loop Structures (if present): If a loop is used to compute multiple π_estimate values (for example, using different n_sides values), the time complexity of the loop will depend on the number of iterations it executes. Assuming the loop runs k times, the time complexity for the loop would be O(k). Total Time Complexity: If π_estimate is calculated only once, the total time complexity is O(1), since all operations have a time complexity of O(1). If a loop is used to calculate multiple π_estimate values, the total time complexity would be O(k), where k is the number of iterations. Relationship with n_sides: It is important to note that the time complexity is not directly related to the value of n_sides. This is because the time complexities of sin, tan, multiplication, and division operations are all O(1). However, as n_sides increases, more loop iterations may be required (if a loop exists) to achieve the same level of accuracy, which could indirectly affect the overall time complexity.