How to Find All the Factors of 24 Step by Step
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 What Is Arc in Maths?
1. What is an arc in math? How do you denote it?
In mathematics, an arc is a portion of the circumference of a circle. It's a curved line segment connecting two points on the circle. To denote an arc, we use the arc symbol (often a curved line above the letters) and the endpoints of the arc. For example, the arc between points P and Q on a circle would be denoted as ↙(PQ). If the arc's length is exactly half the circle's circumference, it's called a semicircular arc.
2. What is the arc length formula in degrees and radians?
The arc length (s) is the distance along the curved portion of the arc. The formulas depend on whether the central angle (θ) is measured in degrees or radians and the circle's radius (r):
- Degrees: s = 2πr (θ/360°)
- Radians: s = rθ
Remember that 2π radians equals 360°. This means that when using radians you are directly relating the arc length to the radius.
3. How do you find the area of a sector formed by an arc?
A sector is the area enclosed by two radii and the arc between them. The formula for the area (A) of a sector is:
- Degrees: A = (πr²)(θ/360°)
- Radians: A = (1/2)r²θ
4. What is a major arc and a minor arc?
An arc can be classified as either a major arc or a minor 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. The sum of the lengths of a major and minor arc is the circle's circumference.
5. What is the relationship between an arc and its corresponding central angle?
The central angle of an arc is the angle formed at the center of the circle by the two radii that define the endpoints of the arc. The measure of the central angle is directly proportional to the length of the arc. A larger central angle corresponds to a longer arc.
6. How are arcs used in calculating the circumference of a circle?
The entire circumference of a circle can be considered a complete arc (360° or 2π radians). Therefore, arc length formulas are fundamental to understanding circumference (C = 2πr). The circumference is the total arc length of the complete circle.
7. Explain the concept of arc measure.
Arc measure refers to the angle subtended by the arc at the center of the circle. It's typically expressed in degrees or radians, and its value directly relates to the arc length.
8. What are some real-world applications of arcs?
Arcs have many practical uses. They appear in:
- Circular motion: Analyzing movement in circular paths (e.g., wheels, gears)
- Engineering: Designing curved structures (e.g., bridges, buildings)
- Mapping: Representing distances and areas on maps
- Trigonometry: Solving problems involving circles and angles
9. How do you find the length of an arc given the radius and central angle in degrees?
Use the formula: s = 2πr (θ/360°), where s is the arc length, r is the radius, and θ is the central angle in degrees.
10. How is the concept of an arc related to chords?
A chord is a straight line segment joining two points on a circle. Every chord defines two arcs: a major arc and a minor arc. The relationship lies in the fact that the chord is a straight line connecting the ends of a curved arc.
11. What is a segment of a circle?
A circular segment is a region bounded by a chord and the arc it subtends. It's the area between a chord and the arc it cuts off from a circle.
12. How is the length of a chord related to the length of the arc it subtends?
For a given arc, the chord is always shorter than the arc length. As the central angle increases, both the arc length and the chord length increase, but the arc length grows faster.
