what is a length of an circular arc
Answers
the distance along the part of the circumference of any circle or any curve (arc). ...
Answer:
Arc length is better defined as the distance along the part of the circumference of any circle or any curve (arc). ... Any distance along the curved line that makes up the arc is known as the arc length. The length of an arc is longer than any straight line distance between its endpoints (a chord).
Step-by-step explanation:
The arc length formula in radians can be expressed as, arc length = θ × r, when θ is in radian. Arc Length = θ × (π/180) × r, where θ is in degree, where,
L = Length of an Arc
θ = Central angle of Arc
r = Radius of the circle
Arc Length Formula in Radians
The arc length of a circle can be calculated using different formulas, based on the unit of the center angle of the arc. The arc length formula in radians can be expressed as,
Arc Length = θ × r
where,
L = Arc Length
θ = Center angle of the arc in radians
r = Radius of the circle
How to Find Arc Length of a Curve?
The arc length of an arc of a circle can be calculated using different methods and formulas based on the given data. Some important cases are given below,
find arc length with the radius and central angle
find arc length without the radius
find arc length without the central angle
How to Find Arc Length With the Radius and Central Angle?
The arc length of a circle can be calculated with the radius and central angle using the arc length formula,
Length of an Arc = θ × r, where θ is in radian.
Length of an Arc = θ × (π/180) × r, where θ is in degree.
How to Find Arc Length Without the Radius?
The arc length of a circle can be calculated without the radius using:
Central angle and the sector area:
Multiply the sector area by 2 and further, divide the result by the central angle in radians.
Find the square root of the result of the division.
Multiply this obtained root by the central angle again to get the arc length.
The units of this calculated arc length will be the square root of the sector area units.
Example: Calculate the arc length of a curve with sector area 25 square units and the central angle as 2 radians.
We have,
Sector area = 25 units
Central angle = 2 radians
Step 1: Sector area × 2 = 25 × 2 = 50
Step 2: 50/central angle = 50/2 = 25
Step 3: √25 = 5
Step 4: 5 × central angle = 5 × 2 = 10 units
Thus, arc length = 10 units
Central angle and the chord length:
Divide the central angle in radians by 2 and further, perform the sine function on it.
Divide the given chord length by twice the result of step 1. This calculation gives you the radius as result.
Multiply the radius by the central angle to get the arc length.
Example: Calculate the arc length of a curve, whose endpoints touch a chord of the circle measuring 5 units. The central angle subtended by the arc is 2 radians.
We have,
Chord length = 5 units
Central angle = 2 radians
Step 1: Central angle/2 = 2/2 = 1
Step 2: Sin(1) = 0.841
Step 3: Chord length/ (2 × 0.841) = 5/ 1.682 = 2.973 units = radius
Step 4: Arc length = radius × central angle = 2.973 × 2 = 5.946 units
Thus, arc length = 5.946 units
How to Find Arc Length Without the Central Angle?
The arc length of a circle can be calculated without the angle using:
Radius and the sector area:
Multiply the sector area by 2.
Then divide the result by the radius squared (the units should be the same) to get the central angle in radians.
Multiply the central angle by the radius to get the arc length.
Example: Calculate the arc length of a curve with sector area 25 square units and radius as 2 units.
We have,
Sector area = 25 units
Central angle = 2 units
Step 1: Sector area × 2 = 25 × 2 = 50
Step 2: 50/radius2 = 50/4 = 12.5 = central angle(rad)
Step 3: Arc length = radius × central angle = 2 × 12.5 = 25 units
Thus, arc length = 25 units
Radius and chord length:
Divide the chord length by twice the given radius.
Find the inverse sine of the obtained result.
Double the result of the inverse sine to get the central angle in radians.
Multiply the central angle by the radius to get the arc length.
Example: Calculate the arc length of a curve, whose endpoints touch a chord of the circle measuring 5 units. The radius of the circle is 2 units.
We have,
Chord length = 5 units
Central angle = 2 units
Step 1: Chord length/(2 × radius) = 5/(2 × 2) = 1.25
Step 2: Sin-1(1.25) = 0.949
Step 3: Central angle = 2 × 0.949 = 1.898 radians
Step 4: Arc length = radius × central angle = 2 × 1.898 = 3.796 units
Thus, arc length = 3.796 units
Important Notes
Given below are key highlights on the concept of arc length.
Arc Length = θ × r, where θ is in radian.
Arc Length = θ × (π/180) × r, where θ is in degree.