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What is the Length of an Arc of a Circle?
Grade Level:
Class 6
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
The length of an arc of a circle is simply the distance along the curved edge of a part of the circle. Imagine cutting a piece of a circular pizza crust; the length of that crust piece is the arc length. It's a fraction of the total circumference of the circle.
Simple Example
Quick Example
Think about a round Ferris wheel at a mela. If you go for a ride and stop halfway, the path you've travelled along the wheel's edge from start to stop is an arc. The length of that path is the arc length.
Worked Example
Step-by-Step
Let's find the length of an arc if the circle has a radius of 7 cm and the arc makes an angle of 90 degrees at the center.
Step 1: Understand the formula. Arc Length = (Angle / 360 degrees) * 2 * pi * Radius.
---Step 2: Identify the given values. Angle = 90 degrees, Radius = 7 cm, pi = 22/7 (approx).
---Step 3: Substitute the values into the formula. Arc Length = (90 / 360) * 2 * (22/7) * 7.
---Step 4: Simplify the fraction (90/360). 90/360 = 1/4.
---Step 5: Multiply the numbers. Arc Length = (1/4) * 2 * (22/7) * 7.
---Step 6: Cancel out common terms. The '7' in the numerator and denominator cancel each other. Arc Length = (1/4) * 2 * 22.
---Step 7: Calculate the final value. Arc Length = (1/4) * 44 = 11 cm.
---The length of the arc is 11 cm.
Why It Matters
Understanding arc length is crucial in fields like engineering and data science. Engineers use it to design curved roads or parts for machines, ensuring they fit perfectly. In computer science, it helps in creating smooth animations or understanding curved data paths.
Common Mistakes
MISTAKE: Using the full circumference instead of a fraction of it. | CORRECTION: Remember that an arc is only a *part* of the circle. You must multiply the circumference by the fraction (angle/360 degrees).
MISTAKE: Confusing radius with diameter. | CORRECTION: The formula uses radius (r), which is half of the diameter (d). If given diameter, divide it by 2 to get the radius before using the formula.
MISTAKE: Forgetting to convert the angle to degrees if it's given in another unit. | CORRECTION: Ensure the angle you use in the formula is always in degrees, as the formula uses '360 degrees' as the total angle in a circle.
Practice Questions
Try It Yourself
QUESTION: A circular park has a radius of 14 meters. What is the length of an arc that makes an angle of 180 degrees at the center? (Use pi = 22/7) | ANSWER: 44 meters
QUESTION: Find the length of an arc of a circle with a radius of 21 cm, if the angle subtended by the arc at the center is 60 degrees. (Use pi = 22/7) | ANSWER: 22 cm
QUESTION: A pizza has a radius of 28 cm. If a slice of pizza is cut such that its crust (arc) length is 44 cm, what is the angle of that slice at the center of the pizza? (Use pi = 22/7) | ANSWER: 90 degrees
MCQ
Quick Quiz
Which of these is the correct formula to find the length of an arc?
(Angle / 360) * pi * Radius
(Angle / 180) * 2 * pi * Radius
(Angle / 360) * 2 * pi * Radius
Angle * 2 * pi * Radius
The Correct Answer Is:
C
Option C is correct because the arc length is a fraction (Angle/360) of the total circumference (2 * pi * Radius). Options A, B, and D use incorrect parts of the formula.
Real World Connection
In the Real World
When ISRO launches satellites, they need to calculate the exact path the satellite will take in its orbit around Earth. This path is often an arc. Calculating its length helps ensure the satellite reaches its target destination accurately. Even in making a round jalebi, the length of the batter's path determines its size!
Key Vocabulary
Key Terms
ARC: A part of the circumference of a circle. | RADIUS: The distance from the center of a circle to any point on its circumference. | CIRCUMFERENCE: The total distance around the edge of a circle. | ANGLE: The measure of the turn between two lines that meet at a point (in this case, at the center of the circle).
What's Next
What to Learn Next
Great job learning about arc length! Next, you can explore how to find the 'Area of a Sector of a Circle'. This builds on arc length because a sector is the pie-slice shape that includes the arc, and understanding one helps understand the other.
