Arc length is the distance along the curved path of a circle or any part of its circumference. We define arc length as measuring the length of a slice of pizza crust. Arc length is calculated using the simple formula: Arc Length= r × θ where 'r' is the radius of the circle and 'θ' is the angle in radians.
In this article, we have covered the Arc Length definition, Arc Length Formula, related examples, and others in detail.
What is Arc Length?
Arc length is a curve or part of the circumference of a circle. It is the distance of the portion of a curve that forms an arc. All the arcs formed are curved in shape and can cover any distance along the direction of the curve.
Arc length is the distance between two points along a section of a curve.
In the image added below in a circle with center O, AB is the circumference of the circle.
Arc length is part of the circumference or distance enclosed between two points on a curve. The two points forming the arc subtends an angle known as the central angle of the arc.
Arc length formula is the formula used for the calculation of the length of an arc. The formula to determine the length of the arc uses the central angle of the arc. These central angles are expressed in the form of radians or degrees. The arc length of a circle is calculated by the product of θ times the radius of the circle.
Mathematically the formula is written as,
Arc Length Formula In Radian:
Arc Length = θ × r
where,
Arc Length Formula In Degree:
Arc Length = θ × (π/180) × r
where,
- θ is the Central Angle Expressed in Degree
- r is the Radius of Circle
| Arc Length Formula (if θ is in degrees) | s = 2 π r (θ/360°) |
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| Arc Length Formula (if θ is in radians) | s = θ × r |
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| Arc Length Formula in Integral Form | s = ∫ ab √{1 + (dy/dx)2} dx |
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How to Find Arc Length Without a Central Angle?
Method 1: The arc length of the circle can be determined by using the radius and sector area of the circle in the condition where the central angle is unknown. The length of the arc without using the central angle can be determined by the given method.
- Step 1: Multiply the sector area of the given circle by 2.
- Step 2: Divide the number by the square of the radius. The central angle will be determined in this step.
- Step 3: Multiply the obtained central angle and the radius of the circle to get the arc length.
Example: Calculate the arc length of a curve sector area 50cm2 and radius measuring as 4cm.
Solution:
Given,
- Sector area is 50 cm2
- Radius is 4 cm
Now, (Sector Area)×2
⇒50×2 = 100
And,
θ = (Arc Length)/r2
⇒100/r2 = 100/(4×4) = 6.25
6.25 is the central angle (In radian).
Then,
Arc Length = (Radius)×(Central Angle)
⇒4×6.25 = 25cm
Method 2: The arc length of the circle can be determined by using the radius and chord length of the circle in the condition where the central angle is unknown. The length of the arc without using the chord length and radius can be determined by the given method.
Step 1: Divide the given chord length by twice the given radius.
Step 2: Calculate the inverse of sine of the number obtained.
Step 3: Multiply the result obtained by the inverse of sine of the number. In this step, the central angle is determined. The central angle obtained is expressed in radian.
Step 4: Multiply the central angle by the radius to determine the arc length.
Example: Calculate the arc length of the curve which touches the chord of length 6cm and subtends a central angle of 2radians.
Solution:
Given,
- Central angle(θ) is 2 radian
- Chord length is 6 cm
Now,
⇒ (central angle)/2
⇒ 2/2=1
And,
⇒ sin(1) = 0.841
And,
⇒ chord length/(2×0.841)
⇒ 6/1.682 = 3.56
This is radius of the circle.
Then, Arc Length = θ × r
⇒ arc length = 2×3.56 = 7.12 cm
Article Related to Arc Length Formula:
Arc Length Formula Solved Examples
Example 1: Find the arc length of a curve on a circle with a radius of 4cm and central angle 2 radians.
Solution:
Given,
- Central angle(θ) is 2 radians
- Radius of circle is 4cm
Now, Arc Length = θ×r
= 2×4
= 8cm
Hence, arc length is 8 cm
Example 2: Find the arc length of a curve on a circle with a radius of 16cm and a central angle of 4radians.
Solution:
Given,
- Central angle(θ) is 4 radians
- Radius of the circle is 16 cm
Now, Arc Length = θ×r
= 4×16
= 64 cm
Hence, the arc length is 8 cm
Example 3: Calculate the arc length of a curve sector area 25 cm2 and radius measuring as 5 cm.
Solution:
Given,
- Sector area is 25 cm2
- Radius is 5 cm
Now,
⇒ (sector area)×2
⇒ 25×2 = 50
And,
θ = (Arc Length)/r2
θ = 50/r2
θ = 50/(5)2
θ = 50/25 = 2
Arc Length = (Radius)×(Central Angle)
= 5×2 = 10cm
Example 4: Calculate the arc length of a curve sector area 40 cm2 and radius measuring as 8 cm.
Solution:
Given,
- Sector area is 40 cm2
- Radius is 8 cm
Now,
Arc Length = (Sector Area)×2
= 40×2 = 80
And,
θ = (Arc Length)/r2
θ = 80/r2
θ = 80/(8×8) = 5/4
Then,
Arc Length = (Radius)×(Central Angle)
⇒ 8×(5/4) = 10cm
Related Articles:
Q1. A sector of a circle has a radius of 10 cm and a central angle of 45∘. Calculate the arc length of the sector.
Q2. Find the arc length of a sector with a central angle of 120∘ and a radius of 7 cm.
Q3. radius of a circle is 15 cm, and the arc length of a sector is 20 cm. Find the central angle of the sector in radians.
Q4. A sector has an area of 30 cm2 and a radius of 5 cm. Calculate the arc length of the sector.
Q5. A circular track has a radius of 50 meters. If an athlete runs along an arc covering an angle of 30∘, what is the distance the athlete runs?
Q6: Find the arc length of a sector with a radius of 20 cm and a central angle of 135 degrees.
Q7: Determine the arc length for a circle with a radius of 25 feet and a central angle of 150 degrees.
Q8: If the radius of a circle is 12 cm and the central angle is 75 degrees what is the length of the arc?
Q9: A sector of a circle has an arc length of 18 cm and a radius of 9 cm. Find the central angle in radians.
Q10: Calculate the arc length of a circle with the radius 30 meters and a central angle of 1.5 radians.
Conclusion
The Calculating the arc length of a circle is essential for the understanding and measuring the curved segments. The formula L=rθ allows for the straightforward computation when the central angle is in the radians with the conversions available for degrees. The Mastery of this concept is crucial in the fields such as the engineering, architecture and various design applications. The Accurate arc length calculations contribute to the precise and effective design solutions. Understanding these principles enhances both the theoretical knowledge and practical application in the real-world scenarios.
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