A circle has a radius of 3 an arc in this circle has a central angle of 20 degrees what is the length of the arc
circumference of whole circle = 2pi r
= 6pi
so :
arc/6pi = 20/360
arc = 120pi/360
= pi/3 = appr 1.047 units
or
arc = rθ , where θ is the angle in radians
so 20 degrees = 20pi/180 radians
= pi/9
arc = 3(pi/9)
= pi/3 , same as above
To find the length of an arc in a circle, we can use the formula:
Arc Length = (θ/360) * 2πr
Where:
- θ is the central angle (in degrees) of the arc
- r is the radius of the circle
Given that the radius of the circle is 3 and the central angle is 20 degrees, we can plug these values into the formula:
Arc Length = (20/360) * 2π(3)
Simplifying this expression:
Arc Length = (1/18) * 2π(3)
Arc Length = (1/9) * π(3)
Arc Length = (π/9)(3)
Arc Length = π/3
So, the length of the arc in this circle is π/3.
To find the length of the arc, you can use the formula:
Arc Length = (Central Angle / 360 degrees) x (2π x Radius)
In this case, the central angle is given as 20 degrees, and the radius is given as 3.
Substituting these values into the formula:
Arc Length = (20 degrees / 360 degrees) x (2π x 3)
To simplify, convert the angle to radians by multiplying by π/180:
Arc Length = (20 degrees x π/180) x (2π x 3)
Arc Length = (20π/180) x (2π x 3)
Now, simplify the equation:
Arc Length = (1/9)π x (6π)
Arc Length = 6π² / 9
Finally, divide by 9 and multiply by π to get the length of the arc:
Arc Length ≈ (6π² / 9) ≈ 6.28319 units (rounded to five decimal places)
Therefore, the length of the arc is approximately 6.28319 units.