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101 LearnersLast updated on September 25, 2025
Math Formula for Central Angle of a Circle
In geometry, understanding the central angle of a circle is crucial for various calculations related to circles. The central angle is the angle subtended at the center of the circle by two given points on the circle. In this topic, we will learn the formula for the central angle of a circle.
List of Math Formulas for Central Angle of a Circle
The central angle is an important concept in circle geometry. Let’s learn the formula to calculate the central angle of a circle.
Math Formula for Central Angle of a Circle
The central angle of a circle is calculated based on the arc length and the radius of the circle.
The formula to calculate the central angle in radians is: Central angle (in radians) = Arc length / Radius
Understanding the Central Angle Formula
The central angle formula relates the arc length and the radius of the circle.
If you know the arc length and the radius, you can find the central angle using the formula:
Central angle = Arc length / Radius
Applications of the Central Angle Formula
The central angle formula is frequently used in various geometry problems and real-world situations involving circles.
For instance, it is used to determine the angular displacement in circular motion or to calculate sectors in pie charts.
Importance of Central Angle Formula
In geometry and real life, we use the central angle formula to analyze and understand circular shapes. Here are some important points about the central angle formula:
It helps in solving problems related to circular arcs and sectors.
By understanding this formula, students can easily grasp concepts related to circle geometry, such as arc length and sector area.
It is a fundamental concept used in fields like engineering, architecture, and astronomy.
Tips and Tricks to Memorize the Central Angle Formula
Students might find geometry formulas tricky and confusing. Here are some tips and tricks to master the central angle formula:
Visualize the circle, its center, and the arc to understand the concept.
Remember the relationship: Central angle = Arc length / Radius.
Use drawings and diagrams to reinforce the formula in your mind.
Common Mistakes and How to Avoid Them While Using the Central Angle Formula
Students make errors when calculating the central angle of a circle. Here are some mistakes and ways to avoid them:
Mistake 1
Confusing arc length with chord length
Students sometimes confuse arc length with chord length. Arc length is the distance along the curved part of the circle, while chord length is the straight line between two points on the circle. Always use arc length in the formula.
Mistake 2
Using incorrect radius
When using the central angle formula, ensure you use the correct radius of the circle. Double-check that the radius is measured from the center of the circle to the arc.
Mistake 3
Incorrect unit conversions
Students sometimes forget to convert units when necessary. Ensure that the arc length and radius are in the same units before applying the formula.
Mistake 4
Misinterpreting radians and degrees
Students often confuse radians with degrees. The central angle formula in radians is direct. If you need degrees, convert radians to degrees by multiplying by 180/π.
Mistake 5
Neglecting to use the formula
Students sometimes try to guess the central angle without using the formula. Always apply the formula: Central angle = Arc length / Radius, for accuracy.
Examples of Problems Using Central Angle of a Circle Formula
Problem 1
Find the central angle when the arc length is 10 units and the radius is 5 units.
The central angle is 2 radians.
Explanation
To find the central angle, use the formula:
Central angle = Arc length / Radius
Central angle = 10 / 5 = 2 radians
Problem 2
A circle has a radius of 7 units. If the arc length is 21 units, what is the central angle?
The central angle is 3 radians.
Explanation
To find the central angle, use the formula:
Central angle = Arc length / Radius
Central angle = 21 / 7 = 3 radians
Problem 3
Calculate the central angle for an arc length of 15 units and a radius of 3 units.
The central angle is 5 radians.
Explanation
To find the central angle, use the formula:
Central angle = Arc length / Radius
Central angle = 15 / 3 = 5 radians
Problem 4
If the central angle is 1 radian and the radius is 4 units, what is the arc length?
The arc length is 4 units.
Explanation
Using the formula:
Arc length = Central angle × Radius
Arc length = 1 × 4 = 4 units
Problem 5
Determine the central angle in degrees for an arc length of 10 units and a radius of 5 units.
The central angle is 114.59 degrees.
Explanation
First, find the central angle in radians:
Central angle = Arc length / Radius
Central angle = 10 / 5 = 2 radians
Convert radians to degrees: 2 × (180/π) ≈ 114.59 degrees
FAQs on Central Angle of a Circle Formula
1.What is the central angle formula?
2.How do you convert the central angle from radians to degrees?
3.What is the arc length in a circle?
4.What is the difference between arc length and chord length?
5.How do you find the arc length if the central angle and radius are known?
Glossary for Central Angle of a Circle Formula
- Central Angle: The angle formed at the center of a circle by two radii.
- Arc Length: The distance measured along the curved line forming part of the circle's circumference.
- Radius: The distance from the center of a circle to any point on its circumference.
- Radians: A unit of angle measure based on the radius of the circle.
- Chord: A straight line connecting two points on a circle.
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Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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: He loves to play the quiz with kids through algebra to make kids love it.
