(a) The Radius Of A Circle Is 5.6 Cm. Find The Angle In Degrees Subtended By An Arc Of 22 Cm At The Centre
Understanding the relationship between the radius of a circle, the length of an arc, and the central angle they subtend is fundamental in geometry. In this problem, we are given the radius of a circle as 5.6 centimeters and an arc length of 22 centimeters. Our goal is to determine the measure of the central angle in degrees that subtends this arc at the center of the circle. This involves applying the basic formula connecting arc length, radius, and central angle, and then converting the resulting measure from radians to degrees for clarity and conventional understanding.
Understanding the Basic Concepts
What Is an Arc?
An arc of a circle is a segment of the circumference — essentially, a "slice" of the circle's boundary. The size of an arc is usually expressed in terms of its length or the angle it subtends at the circle's center.
Central Angle and Arc Length Relationship
The central angle is the angle subtended at the center of the circle by an arc. The length of the arc (L), the radius (r), and the central angle (θ) are mathematically related by the formula:
- Arc Length Formula: L = r × θ (in radians)
This formula is fundamental in solving problems involving circles. It states that the length of an arc is proportional to the radius and the angle it subtends, measured in radians.
Converting Radians to Degrees
Since the problem asks for the angle in degrees, it's important to remember the conversion factor between radians and degrees:
- 1 radian = 180° / π ≈ 57.2958°
Thus, after calculating θ in radians, we will convert it to degrees using this factor for a standard measure.
Step-by-Step Solution
Step 1: Write the Known Values
- Radius of the circle, r = 5.6 cm
- Arc length, L = 22 cm
Step 2: Use the Arc Length Formula
Applying the formula L = r × θ (in radians), we isolate θ:
θ (radians) = L / r
Substituting the known values:
θ (radians) = 22 / 5.6 ≈ 3.9286 radians
Step 3: Convert Radians to Degrees
Using the conversion factor:
θ (degrees) = θ (radians) × (180 / π)
Calculating:
θ (degrees) ≈ 3.9286 × (180 / 3.1416) ≈ 3.9286 × 57.2958 ≈ 225.0°
Step 4: Final Answer
Therefore, the central angle subtended by the arc of length 22 cm at the center of the circle is approximately 225 degrees.
Additional Insights and Validation
Verification of the Result
- Using the calculated angle in radians: θ ≈ 3.9286 radians
- Since the entire circle measures 360°, or 2π radians (~6.2832 radians), the arc subtends a significant portion of the circle:
- Fraction of the circle: θ / 2π ≈ 3.9286 / 6.2832 ≈ 0.625
- This indicates the arc covers approximately 62.5% of the circle, which correlates well with an angle of 225°, since 225° / 360° ≈ 0.625.
Understanding the Context of the Result
Knowing that the arc subtends an angle of approximately 225° at the center implies it is a major arc, covering more than half of the circle. This is consistent with the arc length being nearly four times the radius, which makes sense given the proportionality between arc length and radius.
Conclusion
In conclusion, given a circle with a radius of 5.6 cm and an arc length of 22 cm, the central angle subtended by this arc at the circle's center is approximately 225 degrees. This calculation underscores the importance of understanding the relationship between arc length, radius, and angles, and how conversions between radians and degrees are essential for expressing the final answer in a standard, understandable form.
Summary of Key Points
- The fundamental formula connecting arc length and central angle in radians: L = r × θ.
- To find the angle in radians: θ = L / r.
- Conversion from radians to degrees involves multiplying by 180/π.
- The calculated central angle is approximately 225°, indicating a major arc.
Practical Applications
This type of calculation is essential in various fields such as engineering, astronomy, navigation, and design, where understanding the relationships between angles and arc lengths assists in constructing, analyzing, and interpreting circular structures and motions.
