Problem 14 - Precalculus

Question

If an \(n\) -sided regular polygon is inscribed in a circle of radius \(r\), its perimeter is given by \(P=2 n r \sin \left(\frac{\pi}{n}\right) .\) For a circle with radius \(r=50 \mathrm{mm},\) determine the number of sides needed (to the nearest 25 ) for the perimeter of the polygon to approximate the circumference of the circle correct to two decimal places when rounded.

Step-by-Step Solution

Answer

A polygon with 200 sides approximates the circle's circumference to two decimal places.

Step 1: Understanding the Problem

We need to approximate the circumference of a circle with a regular polygon and find how many sides this polygon should have. The formula for the perimeter of an inscribed polygon is given: \( P = 2 n r \sin\left(\frac{\pi}{n}\right) \) where \( n \) is the number of sides and \( r = 50 \mathrm{mm} \).

Step 2: Formula for Circle's Circumference

The circumference \( C \) of a circle is given by \( C = 2 \pi r \). For the circle in this exercise, the circumference is \( C = 2 \pi \times 50 = 100\pi \). Approximating \( \pi \), we have \( C \approx 314.16 \text{ mm} \) (correct to two decimal places).

Step 3: Equate Perimeter to Circumference

To find the number of sides \( n \), set the polygon perimeter \( P = 314.16 \). Thus, \( 2 n r \sin\left(\frac{\pi}{n}\right) = 314.16 \). Substituting \( r = 50 \), we get \( 100 n \sin\left(\frac{\pi}{n}\right) = 314.16 \).

Step 4: Simplify the Equation

Simplify the equation to \( n \sin\left(\frac{\pi}{n}\right) = 3.1416 \) by dividing both sides by 100.

Step 5: Numerical Iteration

Calculate \( n \sin\left(\frac{\pi}{n}\right) \) for different values of \( n \) to find the closest approximation to 3.1416. Start with \( n = 100 \) and adjust incrementally by 25s to find the optimal \( n \).

Step 6: Check for Closest Value

By testing values: For \( n = 200 \), \( 200 \sin\left(\frac{\pi}{200}\right) \approx 3.1416 \) which closely matches the target value. Adjust as needed.

Key Concepts

Circle Circumference

The circumference of a circle is the distance around it, similar to the perimeter of a polygon. Calculating the circumference is simple when you know the radius. The formula is expressed as \( C = 2 \pi r \), where \( r \) is the radius of the circle. This relationship is based on the constant \( \pi \), approximately 3.14159, which represents the ratio of a circle's circumference to its diameter. In our exercise, for a circle with a radius of 50 mm, the circumference is calculated as \( C = 2 \pi \times 50 \), which simplifies to \( 100 \pi \). When expressed in decimal form, this distance is approximately 314.16 mm. This serves as the benchmark perimeter that our polygon must approximate.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, are fundamental in geometry and are crucial for calculations involving angles and sides of a triangle. In the context of our inscribed polygon problem, the sine function is particularly important. The formula for the perimeter of an inscribed polygon uses the sine of the angle formed by the circle's center and two adjacent vertices of the polygon.

• For a polygon inscribed in a circle, each vertex subtends an angle equal to \( \frac{2\pi}{n} \), where \( n \) is the number of sides.
• Since half of this angle, \( \frac{\pi}{n} \), is used for the sine function, \( \sin\left(\frac{\pi}{n}\right) \) calculates the length of the perpendicular from the center to one of the sides, giving insight into the polygon's geometry.

Understanding this function aids in determining how the polygon's perimeter approaches the circle's circumference as the number of sides increases.

Numerical Iteration

Numerical iteration is a method used to solve equations through trial and error. It is helpful when an exact algebraic solution is not easily obtainable, like in our task of matching the polygon's perimeter to the circle's circumference. The formula \( n \sin\left(\frac{\pi}{n}\right) = 3.1416 \) serves as the equation needing iteration to find the best \( n \).

• Begin by selecting an initial value for \( n \), say 100, then progressively test different values.
• For each \( n \), compute \( n \sin\left(\frac{\pi}{n}\right) \) and compare it to 3.1416.
• Adjust \( n \) incrementally, by 25 in the exercise, to find where the equation is satisfied or the closest match is found.

This step-by-step approach ensures a reasonable approximation is reached, helping to conclude that an \( n = 200 \) provides a close match.