What is Length of an Arc?

S3-SA2-0116

Grade Level:

Class 7

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 part of a circle. Imagine cutting out a slice of a round roti; the length of the arc is the length of the crust of that slice.

Simple Example

Quick Example

Think about a clock face. If the minute hand moves from 12 to 3, it traces out a curved path. The length of this curved path is the length of the arc it covered on the clock's circumference.

Worked Example

Step-by-Step

Problem: A circle has a radius of 7 cm. A sector of this circle has an angle of 90 degrees at the center. Find the length of the arc.

Step 1: Understand the formula. The formula for the length of an arc (L) is L = (theta / 360) * 2 * pi * r, where theta is the central angle in degrees and r is the radius.
---Step 2: Identify the given values. Radius (r) = 7 cm, Central angle (theta) = 90 degrees. Use pi = 22/7.
---Step 3: Substitute the values into the formula: L = (90 / 360) * 2 * (22/7) * 7.
---Step 4: Simplify the fraction (90/360). This simplifies to 1/4.
---Step 5: Calculate: L = (1/4) * 2 * (22/7) * 7.
---Step 6: Cancel out the 7 in the numerator and denominator: L = (1/4) * 2 * 22.
---Step 7: Multiply the remaining numbers: L = (1/4) * 44.
---Step 8: Calculate the final length: L = 11 cm.

Answer: The length of the arc is 11 cm.

Why It Matters

Understanding arc length helps engineers design curved roads or railway tracks, ensuring smooth turns. Data scientists use similar concepts to analyze patterns in circular data. This skill is crucial for careers in architecture, game development, and even satellite navigation.

Common Mistakes

MISTAKE: Using the radius as the diameter in the formula. | CORRECTION: Remember that the formula uses 'r' for radius. If diameter is given, divide it by 2 to get the radius before using the formula.

MISTAKE: Forgetting to divide the central angle by 360 degrees. | CORRECTION: The (theta / 360) part is essential as it tells you what fraction of the whole circle the arc represents.

MISTAKE: Confusing arc length with the area of a sector. | CORRECTION: Arc length is the distance along the curved edge (a 1D measurement), while the area of a sector is the space enclosed by the sector (a 2D measurement).

Practice Questions

Try It Yourself

QUESTION: A circle has a radius of 14 cm. Find the length of an arc that subtends a central angle of 60 degrees. (Use pi = 22/7) | ANSWER: 14.67 cm (approx)

QUESTION: The circumference of a circle is 88 cm. If an arc of this circle has a central angle of 45 degrees, what is its length? | ANSWER: 11 cm

QUESTION: An arc has a length of 11 cm and is part of a circle with a radius of 21 cm. What is the central angle (in degrees) that this arc subtends? (Use pi = 22/7) | ANSWER: 30 degrees

MCQ

Quick Quiz

What does the 'theta' in the arc length formula represent?

The radius of the circle

The diameter of the circle

The central angle of the arc

The circumference of the circle

The Correct Answer Is:

Theta (θ) is always used to represent the central angle of the sector or arc. The other options refer to different parts of the circle or its measurement.

Real World Connection

In the Real World

When you're at a cricket stadium, the boundary ropes often form arcs. Calculating the length of these arcs helps ground staff set up the field correctly. Also, when designing curved slides in water parks, engineers use arc length calculations to ensure the slide is safe and fun.

Key Vocabulary

Key Terms

ARC: A continuous part of the circumference of a circle | RADIUS: The distance from the center of a circle to any point on its circumference | CENTRAL ANGLE: The angle formed at the center of a circle by two radii that connect to the endpoints of an arc | CIRCUMFERENCE: The total distance around the edge of a circle

What's Next

What to Learn Next

Great job understanding arc length! Next, you can explore the 'Area of a Sector'. This concept uses similar ideas but helps you find the space covered by a slice of a circle, which is useful in many real-world problems.