Arc length is the distance measured along a curved line or arc between two points. Unlike a straight-line distance, it follows the curve.
- For a circle, this length is the distance along the curved part of the circle between two points on its circumference.
- If A and B are two points on a circle, then the curved distance measured along the circumference from A to B is called the arc length.
Arc length depends on the following:
- Radius of the circle
- Central angle subtended by the arc
Note: For the same two endpoints: Arc Length > Chord Length, because the curved path is always longer than the straight-line path.
Formula
The formula used to calculate arc length depends on whether the central angle is measured in degrees or radians.
Angle in Degrees
s=\frac{\theta}{360^\circ}\times 2\pi r
- s = Arc length
- r = Radius of the circle
- θ = Central angle in degrees
Angle in Radians
s=r\theta
- (s) = Arc length
- (r) = Radius of the circle
- (θ) = Central angle in radians
Finding Arc Length
Use the steps given below to find the Arc length of the given arc.
Step 1: Note the central angle and length of the radius of the given arc.
Step 2: Use the arc length formula given above according to the value of the angle in degrees or radians.
Step 3: Simplify the above equation to get the required answer.
Proof
Let the radius of circle be r and circumference be C.
The circumference of a circle is: C=2ℼ r
A central angle of (360°) corresponds to the entire circumference.
Using proportionality,
\frac{s}{2\pi r}=\frac{\theta}{360^\circ} Multiplying both sides by (2 ℼ r),
s=\frac{\theta}{360^\circ}(2\pi r) Thus, the arc length formula for angles measured in degrees is obtained.
We know 360° = 2ℼ radian so ,
s=\frac{\theta}{2\pi}(2\pi r) \\[3pts]s= r\theta This gives the arc length formula when the angle is measured in radians.
Solved Examples
Example 1: Find the length of the arc with a radius of 2 m and angle π/2 radians.
Formula to calculate the length of the arc(L) is, L = r θ
Given:
- r = 2 m
- θ = π/2 Radians
Length of Arc = 2 × π/2
Length of Arc = π (π = 3.1415)
Length of Arc = 3.1415 m
Thus, the length of the arc is 3.1415 m.
Example 2: Find the length of the arc with a radius of 5cm and an angle of 60°.
Formula to Calculate the Length of the Arc(L) is, L = 2πr × (θ / 360)
Given:
- r = 5 cm
- θ = 60°
Length of Arc = 2πr × (60 / 360) = 2πr × 1/6 = 2 × 3.1415 × 5/6 (π = 3.1415)
Length of Arc = 5.235 cm
Thus, the length of the arc is 5.235 cm
Example 3: Find the length of the arc with a radius of 10 cm and an angle of 135°.
Formula to calculate the length of the arc(L) is, L = 2πr × (θ / 360)
Given:
- r = 10 cm
- θ = 135°
Length of Arc = 2πr × (135/360) = (2 × 3.1415 × 10 × 135)/360°
Length of Arc = 23.56 cm
Thus, the length of the arc is 23.56 cm.
Practice Problem
Q1. Find the Length of Arc with a radius of 12 cm and an Angle of 60°.
Q2. Find the Radius of Arc with Length of Arc with 26 cm and Angle 60°.
Q3. Find the Length of Arc with a radius of 9 cm and an Angle of 45°.
Q4. Find the Radius of Arc with Length of Arc with 6 cm and Angle 30°.
