Central angle formula properties and how to find it
Understanding the central angle is essential for solving geometry questions in school and competitive exams. It helps you measure arcs, calculate sector areas, and grasp circle theorems—core skills for both CBSE and JEE. Mastering this concept boosts confidence across various maths problems.
Formula Used in Central Angle
The standard formula is: \( \text{Central Angle (in radians)} = \dfrac{\text{Arc Length}}{\text{Radius}} \) or, in degrees, \( \text{Central Angle} = \dfrac{\text{Arc Length} \times 360^\circ}{2\pi \times \text{Radius}} \)
Here’s a helpful table to understand central angle more clearly:
Central Angle Table
| Word | Value | Applies? |
|---|---|---|
| One complete circle | 360° | Yes |
| Semicircle | 180° | Yes |
| Quarter circle | 90° | Yes |
| Reflex angle | More than 180° | Sometimes |
This table shows how the pattern of central angles appears in regular circle divisions and special cases like reflex angles.
Worked Example – Solving a Problem
Let’s solve this:
Find the central angle (in degrees) subtended by an arc of length 11 cm in a circle of radius 7 cm.
1. Write the formula for the central angle in degrees:2. Substitute the values (Arc Length = 11, Radius = 7):
3. Simplify the denominator first:
4. Calculate the angle:
5. Continue calculation:
Final Answer: The central angle is 90°.
Practice Problems
- Calculate the central angle in radians if the arc length is 5 cm and the radius is 4 cm.
- If the central angle of a sector is 120°, what fraction of the full circle is this?
- A circle has an arc of 14 cm and a central angle of 60°. Find the radius.
- Is it possible for a central angle to be 270°? What part of the circle does this represent?
Common Mistakes to Avoid
- Confusing central angle with inscribed angle—central angle’s vertex is always at the circle's center.
- Mixing up degrees and radians in the formula; always check which unit is needed.
- Forgetting to use the correct value of π or miscalculating \(2\pi r\) in the denominator.
Real-World Applications
The concept of central angle is widely used for creating pie charts, designing circular gardens, and engineering curved roads or tracks. Vedantu helps students connect these topics to real-world tasks, making maths more meaningful beyond exams.
We explored the idea of central angle, its formulas, typical questions, and real-life significance. Practice with Vedantu, and use sector of a circle and circle theorems resources to deepen your understanding of central angles in various problems.
FAQs on Central Angle in a Circle Explained Clearly
1. What is a central angle in a circle?
A central angle is an angle whose vertex lies at the center of a circle and whose sides are radii of the circle. It intercepts an arc on the circle.
- The vertex is at the center.
- The two sides are radii.
- The measure of a central angle is equal to the measure of its intercepted arc (in degrees).
2. What is the formula for finding a central angle?
The formula for a central angle in degrees is θ = (Arc Length / Circumference) × 360°.
- Circumference = 2πr
- θ = (s / 2πr) × 360°
3. How do you find the measure of a central angle using arc length?
To find a central angle using arc length, use θ = s / r (in radians).
- Step 1: Identify arc length (s).
- Step 2: Identify radius (r).
- Step 3: Divide s by r.
4. How do you calculate arc length from a central angle?
Arc length is calculated using s = rθ (θ in radians).
- r = radius
- θ = central angle in radians
5. What is the relationship between a central angle and its intercepted arc?
The measure of a central angle is equal to the measure of its intercepted arc in degrees.
- If the central angle is 90°, the arc is 90°.
- This rule applies only when the angle’s vertex is at the center.
6. What is the difference between a central angle and an inscribed angle?
A central angle has its vertex at the center, while an inscribed angle has its vertex on the circle.
- Central angle = equal to intercepted arc.
- Inscribed angle = half of the intercepted arc.
7. How do you find the area of a sector using a central angle?
The area of a sector is given by A = (θ / 360°) × πr² (θ in degrees).
- θ = central angle
- r = radius
8. Can a central angle be greater than 180 degrees?
Yes, a central angle can be greater than 180° and is called a reflex central angle.
- Angles less than 180° form minor arcs.
- Angles greater than 180° form major arcs.
- The maximum central angle in a circle is 360°.
9. How do you convert a central angle from degrees to radians?
To convert a central angle from degrees to radians, use θ (radians) = θ (degrees) × π / 180.
- Multiply the degree measure by π.
- Divide by 180.
10. What are some real-life applications of central angles?
A central angle is used to calculate arc length, sector area, and rotational motion in real-life applications.
- Designing circular tracks and roads.
- Calculating distances on wheels and gears.
- Determining angles in pie charts.
- Engineering and architecture involving circular structures.
