Chapter 1: Circular Measure

Overview

Circular measure provides an alternative to degree measurement for angles using radians. This chapter explores the relationship between degrees and radians, calculations of arc lengths, and areas of sectors and segments. Mastery of circular measure is essential for advanced trigonometry, calculus, and physics applications where angular measurements are required in mathematical calculations.

Learning Objectives

After completing this chapter, you will be able to:

• Convert between degrees and radians
• Calculate arc lengths using radian measure
• Find areas of sectors and segments
• Apply circular measure in real-world problems
• Solve complex geometric problems involving circles

Key Concepts

1.1 Radians

Definition of Radian

A radian is the angle subtended at the center of a circle by an arc equal in length to the radius of the circle.

Key Relationship:

$$\pi \text{ radians} = 180^\circ$$

Conversion Formulas

Degrees to Radians:

$$\text{Radians} = \text{Degrees} \times \frac{\pi}{180^\circ}$$

Radians to Degrees:

$$\text{Degrees} = \text{Radians} \times \frac{180^\circ}{\pi}$$

Common Angle Conversions

1.2 Arc Length and Area of a Sector

Arc Length

For a circle with radius r and central angle θ (in radians):

$$s = r\theta$$

Where:

• s = arc length
• r = radius of circle
• θ = central angle in radians

Area of a Sector

For a circle with radius r and central angle θ (in radians):

$$A = \frac{1}{2}r^2\theta$$

Where:

• A = area of sector
• r = radius of circle
• θ = central angle in radians

Area of a Segment

The area of a segment (area between chord and arc):

$$A_{\text{segment}} = A_{\text{sector}} - A_{\text{triangle}} = \frac{1}{2}r^2\theta - \frac{1}{2}r^2\sin\theta$$ $$A_{\text{segment}} = \frac{1}{2}r^2(\theta - \sin\theta)$$

Important Formulas and Methods

Key Circular Measure Formulas

Problem-Solving Strategies

Arc Length Problems:

1. Convert angle to radians if given in degrees
2. Apply arc length formula s = rθ
3. Calculate and include appropriate units

Sector Area Problems:

1. Convert angle to radians if necessary
2. Apply sector area formula A = ½$r^2$θ
3. Calculate and include square units

Segment Area Problems:

1. Calculate sector area using A = ½$r^2$θ
2. Calculate triangular area using A = ½$r^2$ sin θ
3. Subtract to find segment area

Solved Examples

Example 1: Degree-Radian Conversion

Convert the following angles from degrees to radians: a) 45° b) 120° c) 270°

Solutions:

a) 45° = 45 × (π/180) = π/4 radians

b) 120° = 120 × (π/180) = 2π/3 radians

c) 270° = 270 × (π/180) = 3π/2 radians

Example 2: Radian-Degree Conversion

Convert the following angles from radians to degrees: a) π/3 radians b) 5π/4 radians c) 2π/5 radians

Solutions:

a) π/3 radians = (π/3) × (180/π) = 60°

b) 5π/4 radians = (5π/4) × (180/π) = 225°

c) 2π/5 radians = (2π/5) × (180/π) = 72°

Example 3: Arc Length Calculation

Find the length of an arc that subtends an angle of 60° at the center of a circle with radius 8 cm.

Solution:

First, convert angle to radians: 60° = 60 × (π/180) = π/3 radians

Apply arc length formula: s = rθ = 8 × (π/3) = 8π/3 cm ≈ 8.38 cm

Example 4: Sector Area

Find the area of a sector with radius 10 cm and central angle 150°.

Solution:

First, convert angle to radians: 150° = 150 × (π/180) = 5π/6 radians

Apply sector area formula: A = ½$r^2$θ = ½ × 10² × (5π/6) = ½ × 100 × 5π/6 = 500π/12 = 125π/3 c$m^2$ ≈ 130.90 c$m^2$

Example 5: Segment Area

Find the area of the segment cut off by a chord that subtends an angle of 120° in a circle of radius 6 cm.

Solution:

First, convert angle to radians: 120° = 120 × (π/180) = 2π/3 radians

Calculate sector area: A_sector = ½$r^2$θ = ½ × 6² × (2π/3) = ½ × 36 × 2π/3 = 12π c$m^2$

Calculate triangular area: A_triangle = ½$r^2$ sin θ = ½ × 6² × sin(2π/3) = 18 × (√3/2) = 9√3 c$m^2$

Calculate segment area: A_segment = A_sector - A_triangle = 12π - 9√3 c$m^2$ ≈ 12(3.1416) - 9(1.732) ≈ 37.699 - 15.588 ≈ 22.11 c$m^2$

Example 6: Real-World Application

A Ferris wheel has a diameter of 50 meters. Find: a) The distance traveled by a cabin in one complete revolution b) The area swept by one cabin arm in one revolution

Solution:

Radius r = 50/2 = 25 meters One complete revolution = 360° = 2π radians

a) Arc length (circumference): s = rθ = 25 × 2π = 50π meters ≈ 157.08 meters

b) Sector area (full circle): A = ½$r^2$θ = ½ × 25² × 2π = ½ × 625 × 2π = 625π $m^2$ ≈ 1963.50 $m^2$

Mathematical Derivations

Derivation of Arc Length Formula

For a circle with radius r and central angle θ in radians:

• The full circumference is 2πr
• The full angle is 2π radians
• Therefore, the ratio s/θ = (2πr)/(2π) = r
• Hence: s = rθ

Derivation of Sector Area Formula

For a circle with radius r and central angle θ in radians:

• The full area is π$r^2$
• The full angle is 2π radians
• Therefore, the ratio A/θ = (π$r^2$)/(2π) = $r^2$/2
• Hence: A = ½$r^2$θ

Derivation of Segment Area Formula

The segment area is the difference between sector area and triangular area:

• Sector area: ½$r^2$θ
• Triangular area: ½$r^2$ sin θ
• Therefore: Segment area = ½$r^2$θ - ½$r^2$ sin θ = ½$r^2$(θ - sin θ)

Real-World Applications

1. Engineering and Physics

Rotational Motion:

• Angular velocity in radians per second
• Centripetal acceleration calculations
• Rotational kinetic energy

Example: A wheel with radius 0.3 m rotates at 100 rpm. Find linear speed of a point on the rim.

2. Architecture and Design

Civic Structures:

• Dome and arch calculations
• Circular window designs
• Rotating building elements

Example: Calculate the glass area needed for a circular window with radius 2 m and central angle 120°.

3. Manufacturing

Industrial Applications:

• Gear tooth calculations
• Belt and pulley systems
• Cutting tool paths

Example: Find the length of belt needed to connect two pulleys with radii 10 cm and 15 cm, centers 50 cm apart.

4. Navigation

Aviation and Maritime:

• Course calculations around curves
• Radar scanning areas
• GPS coordinate systems

Example: A radar sweeps through 120° sector with range 50 km. Find the area covered.

Complex Problem-Solving Techniques

Problem: A circular pizza of radius 20 cm is cut into 8 equal slices. Find:

a) The angle subtended by each slice at the center b) The arc length of the crust for each slice c) The area of each slice

Solution:

a) Full circle = 360° = 2π radians Angle per slice = 360°/8 = 45° = 2π/8 = π/4 radians

b) Arc length per slice: s = rθ = 20 × (π/4) = 5π cm ≈ 15.71 cm

c) Area per slice: A = ½$r^2$θ = ½ × 20² × (π/4) = ½ × 400 × π/4 = 50π c$m^2$ ≈ 157.08 c$m^2$

Problem: Find the radius of a circle if an arc of length 12 cm subtends an angle of 72° at the center.

Solution:

First, convert angle to radians: 72° = 72 × (π/180) = 2π/5 radians

Use arc length formula: s = rθ ⇒ 12 = r × (2π/5) ⇒ r = 12 × (5/2π) = 30/π cm ≈ 9.55 cm

Problem: A sector has an area of 50π c$m^2$ and arc length 20 cm. Find the radius and angle.

Solution:

We have two equations:

1. A = ½$r^2$θ = 50π ⇒ $r^2$θ = 100π
2. s = rθ = 20

From equation 2: θ = 20/r

Substitute into equation 1: $r^2$(20/r) = 100π ⇒ 20r = 100π ⇒ r = 100π/20 = 5π cm ≈ 15.71 cm

Then θ = 20/r = 20/(5π) = 4/π radians ≈ 1.27 radians ≈ 72.85°

Problem: Find the area of the segment formed by a chord that subtends an angle of 90° in a circle of radius 8 cm.

Solution:

Convert angle to radians: 90° = 90 × (π/180) = π/2 radians

Calculate sector area: A_sector = ½$r^2$θ = ½ × 8² × (π/2) = ½ × 64 × π/2 = 16π c$m^2$

Calculate triangular area: A_triangle = ½$r^2$ sin θ = ½ × 8² × sin(π/2) = 32 × 1 = 32 c$m^2$

Calculate segment area: A_segment = A_sector - A_triangle = 16π - 32 c$m^2$ ≈ 16(3.1416) - 32 ≈ 50.266 - 32 ≈ 18.27 c$m^2$

Summary Points

• Radian measure is based on circle radius: π radians = 180°
• Arc length formula: s = rθ (θ in radians)
• Sector area formula: A = ½$r^2$θ (θ in radians)
• Segment area formula: A = ½$r^2$(θ - sin θ)
• Conversion between degrees and radians is essential
• Applications span engineering, architecture, manufacturing, and navigation

Common Mistakes to Avoid

1. Angle measurement errors - Always convert to radians for formulas
2. Formula confusion - Use s = rθ for arc length, A = ½$r^2$θ for sector area
3. Unit errors - Ensure consistent units throughout calculations
4. Trigonometric function errors - Use calculator in correct mode (degrees/radians)
5. Sign errors - Pay attention to signs in segment area calculations

SPM Exam Tips

Exam Strategies

1. Master conversions - Practice degree-radian conversions thoroughly
2. Memorize formulas - s = rθ, A = ½$r^2$θ, segment area formula
3. Show conversion steps - Always show degree to radian conversion
4. Include units - Use appropriate units (cm, m, c$m^2$, $m^2$)
5. Check reasonableness - Verify results make geometric sense

Key Exam Topics

• Degree-radian conversions (20% of questions)
• Arc length calculations (25% of questions)
• Sector area calculations (25% of questions)
• Segment area calculations (20% of questions)
• Real-world applications (10% of questions)

Time Management Tips

• Conversion problems: 2-3 minutes
• Arc length problems: 3-4 minutes
• Sector area problems: 3-4 minutes
• Segment area problems: 5-6 minutes
• Application problems: 6-8 minutes

Practice Problems

Level 1: Basic Conversions

1. Convert to radians: a) 30° b) 135° c) 300°

2. Convert to degrees: a) π/6 radians b) 3π/4 radians c) 5π/3 radians

Level 2: Arc Length Calculations

3. Find arc length for: a) r = 10 cm, θ = 45° b) r = 15 cm, θ = 120° c) r = 8 m, θ = 2π/3 radians

4. Find radius if arc length is 12π cm and angle is: a) 60° b) π/3 radians c) 1.5 radians

Level 3: Sector Area Calculations

5. Find sector area for: a) r = 12 cm, θ = 90° b) r = 20 cm, θ = 2π/3 radians c) r = 5 m, θ = 216°

6. Find radius if sector area is 32π c$m^2$ and angle is: a) 45° b) π/2 radians c) 0.8 radians

Level 4: Segment Area Calculations

7. Find segment area for: a) r = 10 cm, θ = 60° b) r = 8 cm, θ = 120° c) r = 6 m, θ = π/3 radians

8. Calculate the area of the segment formed by a chord subtending 150° in a circle of radius 12 cm.

Level 5: Applications

9. Engineering: A pulley with radius 25 cm rotates at 120 rpm. Find: a) Angular velocity in rad/s b) Linear speed of a point on the rim c) Distance traveled in 1 minute

10. Architecture: A circular stained glass window has radius 1.5 m and central angle 100°. Find: a) Arc length of the window edge b) Area of the glass needed c) Area of the segment if a chord cuts off 100°

11. Manufacturing: A circular saw blade has radius 15 cm and cuts through wood at 3000 rpm. Find: a) Angular speed in rad/s b) Linear speed of the teeth c) Distance traveled by a tooth in 1 second

Did You Know? 📚

The radian was formally defined in the 1870s, but the concept dates back to ancient times. The word "radian" was coined by James Thomson in 1873, brother of physicist Lord Kelvin. Radians are mathematically more natural than degrees because they are dimensionless and make calculus operations much simpler. The formula s = rθ works only with radians, which is why higher mathematics exclusively uses radian measure.

Quick Reference Guide

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Circular measure provides the foundation for trigonometry and calculus. Mastering radian measure and the associated formulas will enable you to solve complex problems in physics, engineering, and advanced mathematics with greater precision and elegance.