Arc Length Formula in a Circle with Step by Step Examples
The concept of arc plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding what an arc is, how to calculate arc length, and where arcs appear in geometry is essential for students studying circles, sectors, and other figures.
What Is Arc in Maths?
An arc is a continuous part of the circumference of a circle between two points. If you draw a circle and pick any two points on its outline, the curved segment joining them is called an arc. You’ll find this concept applied in geometry, trigonometry, and also in real-world measurements like designing bridges and roads.
Key Formula for Arc
Here’s the standard formula for measuring the length of an arc:
In degrees: \( s = 2 \pi r \times \frac{\theta}{360} \)
In radians: \( s = r \theta \)
Where:
s = arc length
r = radius of the circle
\(\theta\) = angle subtended at the center (in degrees or radians)
Cross-Disciplinary Usage
Arc is not only useful in Maths but also plays an important role in Physics, Computer Science, and daily logical reasoning. Students preparing for JEE, NEET, or Olympiads will see the relevance of arcs when solving problems on circles, circular motion, sectors, and even algorithms that analyze curves or segments in digital graphics.
Step-by-Step Illustration
- Suppose you have a circle with radius \( r = 7 \) cm, and the angle at the center is \( 90^\circ \). Find the arc length.
Plug into the formula:
\( s = 2 \pi r \times \frac{\theta}{360} \)
- Substitute values:
\( s = 2 \times 3.14 \times 7 \times \frac{90}{360} \)
- Calculate:
\( 2 \times 3.14 \times 7 = 43.96 \)
\( 90/360 = 0.25 \)
Arc length, \( s = 43.96 \times 0.25 = 10.99 \) cm
- Final Answer:
The arc length is 10.99 cm.
Speed Trick or Vedic Shortcut
Here’s a quick shortcut for finding the arc’s fraction of the full circle: If the angle at the center is \( \theta \) degrees, the arc length is just \( \theta/360 \) times the whole circumference. This helps students do mental calculations in geometry exams.
Example Trick: For an arc with a central angle of 60° in a 10 cm radius circle:
- Find circumference: \( 2\pi r = 2 \times 3.14 \times 10 = 62.8 \) cm
- Fraction for arc: \( 60/360 = 1/6 \)
- Multiply: \( 62.8 \times 1/6 = 10.47 \) cm
- So, the arc length = 10.47 cm
Shortcuts like these can save time during Olympiads and class tests. Vedantu’s classes often share such hacks for quicker answers.
Try These Yourself
- Find the arc length of a 45° angle in a circle of radius 14 cm.
- What is the arc if the central angle is 180° in a circle of 8 cm radius?
- Name the arc if end points are Q and R on a circle.
- Is the diameter a type of arc? Explain.
Frequent Errors and Misunderstandings
- Confusing arc with chord (arc is curved, chord is straight).
- Mixing angle units: Always check if angle is in degrees or radians before using the arc formula.
- Forgetting to use the correct value of \( \pi \) as per question instruction.
Relation to Other Concepts
The idea of arc connects closely with topics such as sector of a circle and chord. Mastering arcs makes it much easier to understand the area of a sector and solving questions involving pie diagrams and circular tracks.
Classroom Tip
A handy way to remember arc formulas is: “Arc is just a slice of the circle’s outline—find the slice’s angle, grab the fraction, and multiply with the full circle’s perimeter.” Vedantu’s Maths teachers use coloring activities with pizza slices or clock hands for visual memory!
We explored arc—from definition, formulas for both degrees and radians, shortcuts, worked examples, common mistakes, and links to other Maths ideas. Practice lots of arc questions on Vedantu for CBSE and competitive exams to boost your confidence!
- Sector of a Circle – Learn how arc relates to sector area and angle.
- Chord of Circle – See differences between chords and arcs with diagrams.
- Areas Related to Circles – Explore more about arcs, sectors, and segments.
- Equation of a Circle – Use arc concepts in algebraic circle problems.
FAQs on Arc in Geometry Explained with Meaning and Formula
1. What is an arc in a circle?
An arc is a portion of the circumference of a circle between two points on the circle. It represents a curved segment of the circle’s boundary. In geometry, arcs are named using their endpoints and sometimes a third point to indicate direction. There are two main types of arcs in a circle:
- Minor arc – the shorter arc between two points.
- Major arc – the longer arc between the same two points.
Arcs are closely related to central angles and are commonly used when finding arc length and sector area.
2. What is the formula for arc length?
The arc length formula is s = rθ (when θ is in radians). This formula gives the length of an arc in a circle.
- s = arc length
- r = radius of the circle
- θ = central angle in radians
If the angle is in degrees, use: s = (θ/360°) × 2πr. Always check whether the angle is given in degrees or radians before applying the formula.
3. How do you find the arc length in degrees?
To find arc length when the angle is in degrees, use s = (θ/360°) × 2πr. This formula calculates the fraction of the circle’s circumference.
Steps:
- Identify the radius r.
- Identify the central angle θ in degrees.
- Substitute into (θ/360°) × 2πr.
Example: If r = 7 cm and θ = 90°, then s = (90/360) × 2π × 7 = (1/4) × 14π = 3.5π cm.
4. What is the difference between a minor arc and a major arc?
A minor arc is the shorter arc between two points on a circle, while a major arc is the longer arc between the same two points.
- A minor arc corresponds to a central angle less than 180°.
- A major arc corresponds to a central angle greater than 180°.
Together, a minor arc and major arc between the same endpoints make the full circle of 360°.
5. What is a semicircular arc?
A semicircular arc is an arc that measures 180°, forming exactly half of a circle. It is created when a diameter divides a circle into two equal parts.
- Central angle = 180°
- Arc length = πr
Since the full circumference is 2πr, half of it is πr, which represents the length of a semicircular arc.
6. How do you find the central angle of an arc?
The central angle of an arc can be found using θ = s/r (in radians). This formula rearranges the arc length formula s = rθ.
- θ = central angle in radians
- s = arc length
- r = radius
If working in degrees, use θ = (s / 2πr) × 360°. Always match the unit (degrees or radians) with the formula used.
7. What is the relationship between arc length and radius?
The arc length is directly proportional to the radius when the central angle is fixed. From the formula s = rθ, if θ remains constant, increasing r increases s proportionally.
- If radius doubles, arc length doubles.
- If radius triples, arc length triples.
This shows that larger circles have longer arcs for the same central angle.
8. How do you convert degrees to radians for arc calculations?
To convert degrees to radians, multiply by π/180. This conversion is necessary when using the arc length formula s = rθ.
- Radians = Degrees × π/180
Example: 60° = 60 × (π/180) = π/3 radians. Using radians ensures the formula s = rθ works correctly.
9. What is the formula for the area of a sector using arc angle?
The area of a sector is given by A = (1/2)r²θ when θ is in radians. This formula calculates the portion of the circle enclosed by two radii and an arc.
- A = sector area
- r = radius
- θ = central angle in radians
If θ is in degrees, use: A = (θ/360°) × πr².
10. Can you give an example of finding arc length step by step?
Yes, to find arc length use the formula s = rθ (θ in radians). Here is a worked example.
Example: Find the arc length when r = 5 cm and θ = 2 radians.
- Step 1: Write the formula: s = rθ.
- Step 2: Substitute values: s = 5 × 2.
- Step 3: Multiply to get s = 10 cm.
Therefore, the arc length of the circle is 10 cm.
