This worksheet provides a comprehensive guide to calculating arc length and sector area, crucial concepts in geometry. We'll delve into the formulas, provide step-by-step examples, and offer practice problems to solidify your understanding. Whether you're a student tackling geometry homework or a professional needing a refresher, this resource will equip you with the knowledge and skills to master these calculations.
Understanding Arc Length and Sector Area
Before diving into the calculations, let's define our terms:
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Arc Length: The distance along the curved edge of a circle's sector. Think of it as a portion of the circle's circumference.
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Sector Area: The area enclosed by two radii and an arc of a circle. Imagine a slice of pizza; the sector area is the area of that slice.
Formulas for Arc Length and Sector Area
The calculations rely on understanding the relationship between the arc length/sector area and the circle's full circumference/area. The key is the central angle, denoted as θ (theta), which is the angle subtended at the center of the circle by the arc.
1. Arc Length:
The formula for arc length (s) is:
s = (θ/360°) * 2πr
where:
s= arc lengthθ= central angle in degreesr= radius of the circle2πr= circumference of the circle
2. Sector Area:
The formula for sector area (A) is:
A = (θ/360°) * πr²
where:
A= sector areaθ= central angle in degreesr= radius of the circleπr²= area of the circle
Important Note: These formulas assume θ is measured in degrees. If θ is given in radians, the formulas simplify to:
- Arc Length (radians):
s = rθ - Sector Area (radians):
A = (1/2)r²θ
Step-by-Step Examples
Let's work through a couple of examples to illustrate the application of these formulas.
Example 1: Arc Length
A circle has a radius of 5 cm. Find the arc length of a sector with a central angle of 60°.
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Identify the known variables: r = 5 cm, θ = 60°
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Apply the formula: s = (60°/360°) * 2π(5 cm) = (1/6) * 10π cm ≈ 5.24 cm
Therefore, the arc length is approximately 5.24 cm.
Example 2: Sector Area
A circle has a radius of 10 meters. Find the area of a sector with a central angle of 120°.
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Identify the known variables: r = 10 m, θ = 120°
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Apply the formula: A = (120°/360°) * π(10 m)² = (1/3) * 100π m² ≈ 104.72 m²
Therefore, the sector area is approximately 104.72 m².
Practice Problems
Now it's your turn! Try these problems to test your understanding:
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A circle has a radius of 7 inches. Find the arc length of a sector with a central angle of 45°.
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A circle has a radius of 3 cm. Find the area of a sector with a central angle of 150°.
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A sector has an arc length of 12 cm and a radius of 6 cm. What is the central angle in degrees?
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A sector has an area of 25π square feet and a radius of 10 feet. What is the central angle in degrees?
Solutions (Check Your Answers)
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Approximately 5.49 inches
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Approximately 11.78 square cm
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120 degrees
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90 degrees
This worksheet provides a solid foundation for understanding arc length and sector area calculations. Remember to practice regularly to build confidence and proficiency. Further exploration of related geometric concepts will enhance your overall understanding of circles and their properties.
