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What is the Arc Length Formula (Trigonometric Context)?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
The Arc Length Formula helps us find the distance along the curved edge (arc) of a circle. It tells us how long a 'slice' of the circle's boundary is, based on the circle's radius and the angle that slice makes at the center.
Simple Example
Quick Example
Imagine you have a round pizza, and you cut out one slice. The arc length formula helps you find the length of the crust of that one slice. If the pizza has a radius of 14 cm and your slice makes an angle of 60 degrees, you can find the length of its crust.
Worked Example
Step-by-Step
Let's find the arc length of a sector with a radius of 7 cm and a central angle of 90 degrees.
Step 1: Write down the formula for arc length (L) when the angle is in degrees: L = (theta / 360) * 2 * pi * r. Here, theta is the central angle and r is the radius.
---Step 2: Identify the given values. Radius (r) = 7 cm, Central angle (theta) = 90 degrees.
---Step 3: Substitute the values into the formula: L = (90 / 360) * 2 * pi * 7.
---Step 4: Simplify the fraction: 90 / 360 = 1 / 4.
---Step 5: Calculate: L = (1 / 4) * 2 * (22/7) * 7. (Using pi = 22/7).
---Step 6: Further simplify: L = (1 / 4) * 2 * 22 = 44 / 4.
---Step 7: Final calculation: L = 11 cm.
Answer: The arc length is 11 cm.
Why It Matters
This formula is crucial for designing curved paths in engineering, like roads or railway tracks. In space technology, it helps calculate satellite orbits or the path of a rocket. It's also used in robotics to plan how robot arms move in circular paths.
Common Mistakes
MISTAKE: Using the angle in degrees directly with the formula L = r * theta | CORRECTION: The formula L = r * theta is only for when the angle (theta) is in radians. If the angle is in degrees, use L = (theta / 360) * 2 * pi * r.
MISTAKE: Forgetting to include '2 * pi' or '360' in the formula | CORRECTION: Remember that '2 * pi * r' is the circumference of the whole circle, and 'theta / 360' (or 'theta / 2*pi' for radians) is the fraction of the circle you are considering.
MISTAKE: Confusing arc length with area of a sector | CORRECTION: Arc length measures the distance along the curved edge (like the crust of a pizza slice), while the area of a sector measures the space inside the slice (the cheesy part of the pizza).
Practice Questions
Try It Yourself
QUESTION: A circular park has a radius of 21 meters. What is the length of the arc that subtends an angle of 120 degrees at the center? (Use pi = 22/7) | ANSWER: 44 meters
QUESTION: The minute hand of a clock is 10 cm long. How far does its tip move in 20 minutes? (Hint: The minute hand covers 360 degrees in 60 minutes). (Use pi = 3.14) | ANSWER: 10.47 cm (approx)
QUESTION: An arc of a circle is 22 cm long and the radius of the circle is 14 cm. Find the angle subtended by the arc at the center in degrees. (Use pi = 22/7) | ANSWER: 90 degrees
MCQ
Quick Quiz
What is the arc length of a sector with a radius of 14 cm and a central angle of 45 degrees? (Use pi = 22/7)
5.5 cm
11 cm
22 cm
44 cm
The Correct Answer Is:
B
Using the formula L = (theta / 360) * 2 * pi * r, with theta = 45, r = 14, and pi = 22/7, we get L = (45/360) * 2 * (22/7) * 14 = (1/8) * 2 * 22 * 2 = 11 cm.
Real World Connection
In the Real World
When you see a Ferris wheel at a mela, the path each seat takes from one point to another along the wheel's edge is an arc. Engineers use the arc length formula to design the wheel's structure and ensure each seat travels a consistent, safe distance. Similarly, ISRO scientists use this concept to calculate how much distance a satellite covers in a specific part of its orbit around Earth.
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
What's Next
What to Learn Next
Now that you understand arc length, you can explore the 'Area of a Sector' concept. This will help you calculate the space enclosed by that pizza slice, which also depends on the radius and the central angle. Keep going!
