Calculating Arc Length And Sector Area A Comprehensive Guide
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This article delves into the concepts of arc length and sector area within a circle, providing a step-by-step guide to calculating these values for various central angles. We'll explore the fundamental formulas and apply them to specific examples, enhancing your understanding of circular geometry.
Key Concepts: Arc Length and Sector Area
Before diving into calculations, let's define the key concepts:
- Arc Length: An arc is a portion of the circle's circumference. The arc length is the distance along the curved line of the arc.
- Sector Area: A sector is a region of the circle enclosed by two radii and the arc between them. The sector area is the area of this pie-shaped region.
Formulas for Arc Length and Sector Area
The arc length (L) and sector area (A) can be calculated using the following formulas:
- Arc Length (L) = (θ/360°) × 2πr
- Sector Area (A) = (θ/360°) × πr²
Where:
- θ is the central angle in degrees.
- r is the radius of the circle.
- π (pi) is a mathematical constant approximately equal to 3.14159.
These formulas highlight that the arc length and sector area are directly proportional to the central angle. A larger central angle corresponds to a longer arc and a larger sector area.
Problem Statement: Calculating Arc Length and Sector Area
Consider a circle with center O and circumference of 88 cm. We are given angle AOB = θ° and tasked with finding the length of arc ACB and the area of sector OACB for the following values of θ:
- (a) 60°
- (b) 99°
- (c) 126°
- (d) 216°
This problem provides a practical application of the formulas we discussed earlier. To solve it, we first need to determine the radius of the circle using the given circumference. Then, we can apply the arc length and sector area formulas for each value of θ.
Step 1: Finding the Radius of the Circle
The circumference (C) of a circle is related to its radius (r) by the formula:
- C = 2Ï€r
We are given that the circumference is 88 cm. Substituting this into the formula, we get:
- 88 = 2Ï€r
Solving for r:
- r = 88 / (2Ï€)
- r ≈ 88 / (2 × 3.14159)
- r ≈ 14 cm
Therefore, the radius of the circle is approximately 14 cm. This value will be used in subsequent calculations for arc length and sector area.
(a) θ = 60°: Calculating Arc Length and Sector Area
Now, let's calculate the arc length and sector area for θ = 60°.
Arc Length Calculation (θ = 60°)
Using the arc length formula:
- L = (θ/360°) × 2πr
- L = (60°/360°) × 2 × (22/7) × 14
- L = (1/6) × 2 × (22/7) × 14
- L = (1/3) × (22/7) × 14
- L = (1/3) × 22 × 2
- L = 44/3
- L ≈ 14.67 cm
Therefore, the length of arc ACB when θ = 60° is approximately 14.67 cm.
Sector Area Calculation (θ = 60°)
Using the sector area formula:
- A = (θ/360°) × πr²
- A = (60°/360°) × (22/7) × 14 × 14
- A = (1/6) × (22/7) × 14 × 14
- A = (1/3) × (11/7) × 14 × 14
- A = (1/3) × 11 × 2 × 14
- A = (1/3) × 308
- A ≈ 102.67 cm²
Thus, the area of sector OACB when θ = 60° is approximately 102.67 cm².
(b) θ = 99°: Calculating Arc Length and Sector Area
Next, we calculate the arc length and sector area for θ = 99°.
Arc Length Calculation (θ = 99°)
Using the arc length formula:
- L = (θ/360°) × 2πr
- L = (99°/360°) × 2 × (22/7) × 14
- L = (11/40) × 2 × (22/7) × 14
- L = (11/20) × (22/7) × 14
- L = (11/10) × (22/7) × 7
- L = (11/10) × 22
- L = 242/10
- L = 24.2 cm
Therefore, the length of arc ACB when θ = 99° is 24.2 cm.
Sector Area Calculation (θ = 99°)
Using the sector area formula:
- A = (θ/360°) × πr²
- A = (99°/360°) × (22/7) × 14 × 14
- A = (11/40) × (22/7) × 14 × 14
- A = (11/20) × (22/7) × 7 × 14
- A = (11/10) × (11/1) × 14
- A = (11/5) × 11 × 7
- A = 847/5
- A = 169.4 cm²
Hence, the area of sector OACB when θ = 99° is 169.4 cm².
(c) θ = 126°: Calculating Arc Length and Sector Area
Let's proceed to calculate the arc length and sector area for θ = 126°.
Arc Length Calculation (θ = 126°)
Using the arc length formula:
- L = (θ/360°) × 2πr
- L = (126°/360°) × 2 × (22/7) × 14
- L = (7/20) × 2 × (22/7) × 14
- L = (7/10) × (22/7) × 14
- L = (1/5) × 11 × 2
- L = 44/5
- L = 30.8 cm
Therefore, the length of arc ACB when θ = 126° is 30.8 cm.
Sector Area Calculation (θ = 126°)
Using the sector area formula:
- A = (θ/360°) × πr²
- A = (126°/360°) × (22/7) × 14 × 14
- A = (7/20) × (22/7) × 14 × 14
- A = (7/10) × (11/1) × 14
- A = (7/5) × 11 × 7
- A = 1078/5
- A = 215.6 cm²
Thus, the area of sector OACB when θ = 126° is 215.6 cm².
(d) θ = 216°: Calculating Arc Length and Sector Area
Finally, we calculate the arc length and sector area for θ = 216°.
Arc Length Calculation (θ = 216°)
Using the arc length formula:
- L = (θ/360°) × 2πr
- L = (216°/360°) × 2 × (22/7) × 14
- L = (6/10) × 2 × (22/7) × 14
- L = (3/5) × 2 × (22/7) × 14
- L = (3/5) × 2 × 22 × 2
- L = (3/5) × 88
- L = 264/5
- L = 52.8 cm
Therefore, the length of arc ACB when θ = 216° is 52.8 cm.
Sector Area Calculation (θ = 216°)
Using the sector area formula:
- A = (θ/360°) × πr²
- A = (216°/360°) × (22/7) × 14 × 14
- A = (6/10) × (22/7) × 14 × 14
- A = (3/5) × (22/7) × 14 × 14
- A = (3/5) × 22 × 2 × 14
- A = (3/5) × 616
- A = 1848/5
- A = 369.6 cm²
Hence, the area of sector OACB when θ = 216° is 369.6 cm².
Summary of Results
Here's a summary of the arc lengths and sector areas calculated for each value of θ:
| θ (degrees) | Arc Length (cm) | Sector Area (cm²) |
|---|---|---|
| 60 | 14.67 | 102.67 |
| 99 | 24.2 | 169.4 |
| 126 | 30.8 | 215.6 |
| 216 | 52.8 | 369.6 |
Conclusion
This article has demonstrated how to calculate arc length and sector area of a circle given the central angle and circumference. By applying the formulas L = (θ/360°) × 2πr and A = (θ/360°) × πr², we successfully determined these values for various central angles. Understanding these concepts is crucial for various applications in geometry, engineering, and other fields.
By mastering these calculations, you gain a deeper appreciation for the relationship between angles, arc lengths, and sector areas within a circle. This knowledge empowers you to solve a wide range of geometric problems and apply these principles in real-world scenarios.