Lesson 3: Arc Length and Area of a Sector
Estimated time: 25-30 minutes
Learning Objectives
By the end of this lesson, you will be able to:
- Derive the arc length formula from the definition of a radian
- Calculate arc length given a central angle and radius
- Derive and apply the sector area formula
- Solve real-world problems involving arc length and sector area
Arc Length Formula
Recall that a radian is defined so that θ = s/r, where s is the arc length and r is the radius. Rearranging gives us the arc length formula.
Arc Length Formula
s = rθ
where s is the arc length, r is the radius, and θ must be in radians.
This formula follows directly from the radian definition. The fraction of the full circumference (2πr) that the arc represents equals the fraction of the full rotation (2π) that θ represents:
s / (2πr) = θ / (2π), so s = rθ.
Example 1: Finding Arc Length
Find the arc length intercepted by a central angle of π/4 radians in a circle of radius 10 cm.
Solution:
Step 1: Identify the values: r = 10, θ = π/4
Step 2: Apply the formula: s = rθ = 10 × π/4 = 10π/4 = 5π/2
Step 3: Approximate: 5π/2 ≈ 7.85 cm
Example 2: Finding Arc Length from Degrees
Find the arc length intercepted by a 60° angle in a circle of radius 12 inches.
Solution:
Step 1: Convert to radians: 60° = π/3
Step 2: Apply the formula: s = 12 × π/3 = 4π
Step 3: Approximate: 4π ≈ 12.57 inches
Example 3: Finding the Angle
An arc of length 15 cm is intercepted by a central angle in a circle of radius 6 cm. Find the angle in radians.
Solution:
θ = s / r = 15 / 6 = 5/2 = 2.5 radians
Area of a Sector
A sector is the region bounded by two radii and the intercepted arc — like a slice of pie.
Area of a Sector
A = (1/2) r² θ
where r is the radius and θ must be in radians.
This formula comes from the proportion: the sector area is to the full circle area as θ is to 2π:
A / (πr²) = θ / (2π), so A = (1/2)r²θ.
Example 4: Finding Sector Area
Find the area of a sector with radius 8 m and central angle 2π/3 radians.
Solution:
Step 1: Identify: r = 8, θ = 2π/3
Step 2: A = (1/2)(8)²(2π/3) = (1/2)(64)(2π/3) = 64π/3
Step 3: Approximate: 64π/3 ≈ 67.02 m²
Example 5: Sector Area from Degrees
Find the area of a sector with radius 5 ft and central angle of 72°.
Solution:
Step 1: Convert: 72° = 72 × π/180 = 2π/5
Step 2: A = (1/2)(5)²(2π/5) = (1/2)(25)(2π/5) = 25π/5 = 5π
Step 3: Approximate: 5π ≈ 15.71 ft²
Finding Unknown Quantities
Both formulas can be rearranged to solve for any of the three variables (s or A, r, θ).
Example 6: Finding the Radius
A sector has an arc length of 18 cm and a central angle of 3 radians. Find the radius.
Solution:
s = rθ, so r = s / θ = 18 / 3 = 6 cm
Example 7: Finding the Angle from Area
A sector has area 50 in² and radius 10 in. Find the central angle.
Solution:
A = (1/2)r²θ, so θ = 2A / r² = 2(50) / 100 = 1 radian
Real-World Applications
Example 8: Windshield Wiper
A windshield wiper is 20 inches long and sweeps through an angle of 110°. What area of the windshield does it clean?
Solution:
Step 1: Convert: 110° = 110π/180 = 11π/18
Step 2: A = (1/2)(20)²(11π/18) = (1/2)(400)(11π/18) = 2200π/18 = 1100π/9
Step 3: Approximate: 1100π/9 ≈ 383.97 in²
Example 9: Latitude and Distance
Two cities on the same meridian are separated by 5° of latitude. Using Earth's radius of 3960 miles, find the distance between them along Earth's surface.
Solution:
Step 1: Convert: 5° = 5π/180 = π/36
Step 2: s = rθ = 3960 × π/36 = 110π
Step 3: Approximate: 110π ≈ 345.6 miles
Common Mistakes to Avoid
- Using degrees in the formula: s = rθ and A = (1/2)r²θ require θ in radians. Always convert first!
- Forgetting the 1/2: The sector area formula has a factor of 1/2. Do not confuse it with s = rθ.
- Mixing units: If the radius is in centimeters, the arc length will be in centimeters and the area in square centimeters.
Check Your Understanding
1. Find the arc length intercepted by a central angle of 2 radians in a circle of radius 7 cm.
2. Find the area of a sector with radius 9 m and central angle π/6.
3. An arc is 20 inches long on a circle of radius 8 inches. What is the central angle in radians?
Key Takeaways
- Arc length: s = rθ (with θ in radians).
- Sector area: A = (1/2)r²θ (with θ in radians).
- Both formulas can be rearranged to solve for r or θ.
- Always convert degrees to radians before substituting into these formulas.
- These formulas are used extensively in physics, engineering, and navigation.
Ready for More?
Next Lesson
In Lesson 4 you will learn about angular and linear speed and how they relate to each other.
Start Lesson 4