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What is the Length of an Arc?
Grade Level:
Class 6
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
The length of an arc is the distance along the curved edge of a circle or a part of a circle. Imagine cutting out a piece of the circle's edge and straightening it; its length would be the arc length.
Simple Example
Quick Example
Think about a round plate of dosa. If you cut out a small curved piece from its edge, the length of that curved piece is an arc length. It's just a part of the total boundary (circumference) of the dosa.
Worked Example
Step-by-Step
PROBLEM: A circle has a radius of 7 cm. What is the length of an arc that makes a 90-degree angle at the center? (Use pi = 22/7)
STEP 1: Understand the formula for arc length. Arc Length = (Angle / 360 degrees) * 2 * pi * Radius.
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STEP 2: Identify the given values. Angle = 90 degrees, Radius = 7 cm, pi = 22/7.
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STEP 3: Substitute the values into the formula. Arc Length = (90 / 360) * 2 * (22/7) * 7.
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STEP 4: Simplify the fraction (90/360). 90/360 = 1/4.
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STEP 5: Substitute the simplified fraction. Arc Length = (1/4) * 2 * (22/7) * 7.
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STEP 6: Cancel out common terms. The '7' in the numerator and denominator cancel out. Arc Length = (1/4) * 2 * 22.
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STEP 7: Multiply the remaining numbers. Arc Length = (1/4) * 44 = 11.
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ANSWER: The length of the arc is 11 cm.
Why It Matters
Understanding arc length is crucial for designing curved structures and paths, like roads or bridges. Engineers use this concept to calculate materials needed, and even in computer science, it helps in creating smooth animations and graphics for games.
Common Mistakes
MISTAKE: Using the area formula instead of the circumference formula. | CORRECTION: Remember, arc length is a distance along the edge, so it relates to circumference (2 * pi * r), not area (pi * r^2).
MISTAKE: Forgetting to divide the central angle by 360 degrees. | CORRECTION: The angle given is only a *part* of a full circle, so you must find what *fraction* of the circle the arc represents by dividing the angle by 360.
MISTAKE: Mixing up radius and diameter. | CORRECTION: The formula uses radius (r). If diameter (d) is given, remember that radius = diameter / 2.
Practice Questions
Try It Yourself
QUESTION: A circle has a radius of 14 cm. Find the length of an arc that subtends an angle of 60 degrees at the center. (Use pi = 22/7) | ANSWER: 14.67 cm (approx)
QUESTION: The circumference of a circle is 88 cm. What is the length of an arc that forms a 45-degree angle at the center? | ANSWER: 11 cm
QUESTION: An arc length is 16.5 cm and it makes an angle of 180 degrees at the center of a circle. What is the radius of this circle? (Use pi = 22/7) | ANSWER: 5.25 cm
MCQ
Quick Quiz
Which formula correctly represents the length of an arc?
(Angle / 360) * pi * r^2
(Angle / 360) * 2 * pi * r
Angle * 2 * pi * r
2 * pi * r
The Correct Answer Is:
B
Option B is correct because arc length is a fraction of the circle's circumference (2 * pi * r), and that fraction is determined by the central angle divided by 360 degrees. Option A is for area, and C and D are incomplete or incorrect.
Real World Connection
In the Real World
When ISRO launches a satellite, its path around the Earth can be thought of as an arc. Calculating the length of this arc helps scientists determine how far the satellite travels in a certain time, which is important for communication and navigation systems like GPS.
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 | CENTRAL ANGLE: The angle formed at the center of a circle by two radii that meet the endpoints of an arc
What's Next
What to Learn Next
Now that you understand arc length, you can learn about the 'Area of a Sector'. A sector is like a slice of pizza, and knowing arc length will help you understand its curved boundary and how it relates to the entire circle's area.
