Section 11.1: Circumference and Arc Length Detailed Study Guide

Core Concepts of Circumference

• Definition of Circumference: The circumference of a circle ($C$) is the distance around the circle.
• Geometric Derivation: If a regular polygon is inscribed in a circle, as the number of sides increases, the polygon approximates the circle. The ratio of the perimeter of the polygon to the diameter of the circle approaches the mathematical constant $\pi \approx 3.14159…$
• The Ratio $\pi$: For all circles, the ratio of the circumference $C$ to the diameter $d$ is the same: $\frac{C}{d} = \pi$.
• Circumference Formulas:     * $C = \pi d$     * $C = 2\pi r$ (since $d = 2r$)
• Precision and Rounding: Throughout the study of circumference, area, and volume, use the $\pi$ key on a calculator for calculations and round the final answer to the nearest hundredths place unless otherwise specified.

Examples: Using Circumference Formulas

• Finding Circumference from Radius: Given a circle with radius $r = 9\text{ centimeters}$     * $C = 2\pi r = 2 \times \pi \times 9 = 18\pi$     * $C \approx 56.55\text{ centimeters}$
• Finding Radius from Circumference: Given a circle with circumference $C = 26\text{ meters}$     * $26 = 2\pi r$     * $r = \frac{26}{2\pi} \approx 4.14\text{ meters}$
• Monitoring Progress Exercises:     * Circle with diameter $d = 5\text{ inches}$: $C = 5\pi \approx 15.71\text{ inches}$.     * Circle with circumference $C = 17\text{ feet}$: $d = \frac{17}{\pi} \approx 5.41\text{ feet}$.

Arc Length Concepts and Formulas

• Definition of Arc Length: An arc length is a portion of the circumference of a circle. While the measure of an arc is in degrees, the length of an arc is measured in linear units (e.g., centimeters, inches).
• Arc Length Ratio: In a circle, the ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to $360^{\circ}$.
• Arc Length Formulas:     * $\frac{\text{Arc length of } \overset{\frown}{AB}}{2\pi r} = \frac{m \overset{\frown}{AB}}{360^{\circ}}$     * $\text{Arc length of } \overset{\frown}{AB} = \frac{m \overset{\frown}{AB}}{360^{\circ}} \times 2\pi r$

Examples: Finding Arc Length and Measures

• Finding Arc Length: Circle with radius $r = 8\text{ cm}$ and arc measure $m \overset{\frown}{AB} = 60^{\circ}$.     * $\text{Arc length of } \overset{\frown}{AB} = \frac{60^{\circ}}{360^{\circ}} \times 2\pi(8) = \frac{1}{6} \times 16\pi \approx 8.38\text{ cm}$
• Finding Circumference from Arc Length: Circle $Z$ where arc $XY$ has measure $40^{\circ}$ and length $4.19\text{ in}$.     * $\frac{4.19}{C} = \frac{40^{\circ}}{360^{\circ}} = \frac{1}{9}$     * $C = 4.19 \times 9 = 37.71\text{ in}$
• Finding Arc Measure from Length: Circle with radius $r = 15.28\text{ m}$ and arc length $RS = 44\text{ m}$.     * $\frac{44}{2\pi(15.28)} = \frac{m \overset{\frown}{RS}}{360^{\circ}}$     * $m \overset{\frown}{RS} = 360^{\circ} \times \frac{44}{2\pi(15.28)} \approx 165^{\circ}$

Real-Life Applications

• Distance Traveled by a Tire:     * A tire has a rim diameter of $15\text{ in}$ and sidewalls of $5.5\text{ in}$ on each side. The total diameter $d = 15 + 2(5.5) = 26\text{ in}$.     * Circumference $C = 26\pi\text{ in}$.     * Distance in 15 revolutions: $15 \times 26\pi \approx 1225.2\text{ in}$.     * Conversion to feet: $\frac{1225.2}{12} \approx 102.1\text{ feet}$.
• Track Running Distance:     * A track consists of two straight sections of $84.39\text{ m}$ and two $180^{\circ}$ arcs (semicircles).     * Inner path radius $r = 36.8\text{ m}$. Total distance: $2(84.39) + 2(\frac{1}{2} \times 2\pi \times 36.8) \approx 400.0\text{ meters}$.     * Outer path (blue path) radius $r = 44.02\text{ m}$. Total distance: $2(84.39) + 2(\frac{1}{2} \times 2\pi \times 44.02) \approx 445.4\text{ meters}$.
• Motorcycle Skills Test:     * Diameter of tire = $25\text{ inches}$.     * If the tire makes exactly one-half additional revolution: Distance = $\frac{1}{2} \times \pi \times 25 \approx 39.27\text{ inches}$.

Measuring Angles in Radians

• Definition of Radian: The radian measure of a central angle is defined as the constant of proportionality between the arc length and the radius. In a circle of radius 1, the radian measure is equal to the length of the arc associated with the angle.
• Full Circle Measure: A complete circle ($360^{\circ}$) is exactly $2\pi\text{ radians}$.
• Conversion Formulas:     * Degrees to Radians: Multiply degree measure by $\frac{\pi \text{ radians}}{180^{\circ}}$.     * Radians to Degrees: Multiply radian measure by $\frac{180^{\circ}}{\pi \text{ radians}}$.
• Conversion Examples:     * Convert $45^{\circ}$ to radians: $45^{\circ} \times \frac{\pi}{180^{\circ}} = \frac{\pi}{4}\text{ radian}$.     * Convert $\frac{3\pi}{2}\text{ radians}$ to degrees: $\frac{3\pi}{2} \times \frac{180^{\circ}}{\pi} = 270^{\circ}$.     * Convert $15^{\circ}$ to radians: $15^{\circ} \times \frac{\pi}{180^{\circ}} = \frac{\pi}{12}\text{ radian}$.     * Convert $\frac{4\pi}{3}$ to degrees: $\frac{4\pi}{3} \times \frac{180^{\circ}}{\pi} = 240^{\circ}$.

Historical Case Study: Eratosthenes’ Earth Circumference Estimation

• Method: Over 2000 years ago, Greek scholar Eratosthenes estimated the Earth's circumference by observing sunlight in two cities: Syene and Alexandria.
• Data Points:     * In Syene, the Sun shone straight down a well (vertical).     * In Alexandria, a vertical stick cast a shadow, indicating the Sun's rays made a $7.2^{\circ}$ angle with the stick.     * The distance between the two cities was approximately $575\text{ miles}$.
• Logic: Because the Sun's rays are parallel, the angle at the Earth's center is also $7.2^{\circ}$. This central angle represents a fraction of the Earth's circumference.
• Calculation:     * $\frac{7.2^{\circ}}{360^{\circ}} = \frac{575}{\text{Circumference}}$     * $\text{Circumference} = \frac{575 \times 360}{7.2} = 28750\text{ miles}$.

Advanced Problems and Conceptual Applications

• London Eye: A Ferris wheel traveling at $0.26\text{ m/s}$ with a diameter of roughly $135\text{ m}$ (implied radius $67.5\text{ m}$). The circumference is $C = 135\pi \approx 424.12\text{ m}$. Time for one revolution: $\frac{424.12}{0.26} \approx 1631\text{ seconds} \approx 27.2\text{ minutes}$.
• Mathematical Connections (Problem 39): Circles A, B, and C have radii $x, 3x, \text{ and } 5x$. Sum of circumferences = $63\pi$.     * $2\pi(x) + 2\pi(3x) + 2\pi(5x) = 63\pi$     * $2\pi(9x) = 18\pi x = 63\pi$     * $x = \frac{63}{18} = 3.5$.     * Segment $AC$ traverses the diameters of A and B plus the radius of C or specific segments defined by the visual diagram ($AC = x + 2(3x) + 5x = 12x = 42$?).
• Clock Hand Angles:     * 1:30 p.m.: The minute hand is at $180^{\circ}$. The hour hand is halfway between 1 and 2 ($45^{\circ}$ from vertical). The angle is $135^{\circ}$ or $\frac{3\pi}{4}\text{ radians}$.     * 3:15 p.m.: The minute hand is at $90^{\circ}$. The hour hand is 1/4 of the way between 3 and 4. Since each hour is $30^{\circ}$, the hour hand is moved $7.5^{\circ}$. The angle is $7.5^{\circ}$ or $\frac{\pi}{24}\text{ radians}$.
• Gears: When a large gear (radius 7) turns, the smaller gear (radius 3) completes revolutions in a ratio of $7:3$. One revolution of the large gear = $\frac{7}{3} \approx 2.33$ revolutions of the smaller gear.
• Rational Approximation of $\pi$: $\pi$ is irrational. The fraction $\frac{355}{113}$ is a common rational approximation ($\approx 3.1415929$) that matches $\pi$ to six decimal places.