How Does a Central Angle Relate to Its Arc?
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 What Is a Central Angle? Meaning, Properties, and Examples
1. What is a central angle?
Central angle is an angle whose vertex is at the center of a circle and whose sides (arms) extend to the circumference, intersecting the circle in two points. The arc between these two points is called the intercepted arc of the central angle.
2. What is a central angle best described as?
A central angle is best described as an angle that is formed at the center of a circle by two radii. It measures the angle between two radii drawn from the center to the circumference of the circle. This angle directly subtends an arc on the circle.
3. Is the central angle half the arc?
No, the central angle is not half the arc. The measure of a central angle in degrees is equal to the degree measure of its intercepted arc. For example, if the arc measures 60°, the central angle is also 60°.
4. What is the central angle of 4?
The phrase 'central angle of 4' is incomplete. If you mean a central angle that intercepts an arc length of 4 units, you need the circle’s radius to calculate the angle. Use the formula: Central Angle (in radians) = Arc Length / Radius.
5. What is the central angle of a circle?
The central angle of a circle is an angle whose vertex is at the circle’s center and whose sides (arms) pass through two points on the circumference, making an arc between those points. The sum of all central angles in a circle is 360°.
6. What is the central angle definition in geometry?
In geometry, a central angle is defined as an angle with its vertex at the center of the circle and its sides (radii) intersecting the circle at two different points. This angle subtends an arc on the circle.
7. What is the central angle formula?
Central angle formula:
If you know the arc length (L) and the radius (r) of a circle, the formula is:
Central Angle (θ in radians) = L / r
To convert radians to degrees: θ (in degrees) = (L / r) × (180/π).
8. What is the central angle theorem?
The central angle theorem states that the measure of a central angle is equal to the measure of its intercepted arc. In contrast, an inscribed angle subtending the same arc is always half the central angle.
9. What is the relationship between central angles and arc measures?
The measure of a central angle in degrees is equal to the arc measure of the arc it intercepts. For example, a 90° central angle intercepts a 90° arc of the circle.
10. What is the difference between a central angle and an inscribed angle?
A central angle's vertex is at the center of the circle, while an inscribed angle's vertex is on the circumference. The degree measure of an inscribed angle is half the measure of its intercepted arc, while for a central angle, it is exactly the same as the arc measure.
11. How do you calculate the central angle of a sector in degrees?
To find the central angle of a sector in degrees, use:
Central Angle = (Area of Sector / Total Area of Circle) × 360°
Or if you have arc length: Central Angle = (Arc Length / Circumference) × 360°
12. What is the central angle formula for a pie chart?
For pie charts, the central angle for a data category is:
Central Angle = (Category Value / Total Value) × 360°
This formula helps in dividing a pie chart accurately based on collected data.
