The arc length formula is used to calculate and find the length of an arc of a circle. In mathematics, a smooth curve joining two points is known as an arc. Any distance that moves along the curved line that makes up an arc is regarded as the arc length. For a circle, the arc length formula is given as, s = rθ, where s is the arc length in radians, r is the radius of the circle, and θ is the central angle in radians. This seems to be quite complicated but trust me if you understand and grasp some basic concepts about arc length formulas, you can master any problems regarding this topic. In this session, we will try to learn about the arc length formula with radians, how to find arc length with and without the radius and see some related examples.
Arc Length Formula with Radians
The length of an arc can be calculated using various formulas, based on the unit of the central angle of the arc. The arc length formula of a circle can be given as s = rθ when θ is in radian
where,
S = the arc length
r = the radius of the circle
θ = the central angle of the arc
Let us see some examples to understand the arc length formula in a better way,
- Calculate the length of an arc if the diameter of the circle is 16 cm and the central angle is 40 degrees.
According to the question,
Diameter = 16 cm
Radius = 16/2 = 8 cm
The central angle of the circle is = 40 degrees
We know that arc length formula is = rθ
Now putting the values of r and θ, 4.40 degree / 360 degree.
The answer comes to 5.582 cm.
Thus, the arc length of the circle is 5.582cm.
- Find the length of an arc of a circle that provides an angle of 120 degrees to the centre of the circle with 24 cm.
According to the question,
The radius of the circle is = 24 cm
The central angle of the circle is = 120 degrees
We know that the arc length formula is = r θ
Now putting the values of r and θ,
The answer comes to 50.24 cm
Thus, the arc length of the circles is 50.24 cm.
How to Find the Arc Length of a Circle Without Using the Radius?
The following points mentioned below shows us how can we calculate the arc length of the circle without using the radius:
Central angle and the sector area:
1. We can multiply the sector area by 2 and then divide the answer by the central angle in radians.
2. Then, find the square root of the result of the division.
3. Then, multiply the obtained root by the central angle again to get the arc length.
4. The final result should be written in units.
Let us take an example to grasp this concept in a better way:
Calculate the arc length of a curve with sector area 25 square units and the central angle as 2 radians:
Step1: 25 × 2 = 50
Step2: 50 / 2 = 25
Step3: √25 = 5
Step4: 5 multiplied by central angle = 5 × 2 = 10 units.
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