A part of a curve or a part of a one of a one is referred to as Arc.All of lock havea curvein their shape. The curved section of this objects is mathematically called an arc. Arclength is better defined as the distance follow me the part of the one of any circle or any curve (arc). Any kind of distance along the bent line that provides up the arc is well-known as the arc length. The length of one arc is longer than any kind of straight heat distance in between its endpoints (a chord).

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1.What is Arc Length?
2.Arc size Formula
3.How to uncover Arc length of a Curve?
4.FAQs top top Arc Length

What isArc Length?

Thearc length is defined as the interspace in between thetwo points along a section of a curve. Anarcof acircleis any part of the circumference. Theangle subtended by an arcat any allude is the edge formed between the two line segments joining that allude to the end-points the the arc. For example, in the circle shown below, OP is the arc that the one with facility Q. The arc length of this arc OP is provided as L.

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Arc size Formula


The length of an arc can be calculate using various formulas, based on the unit that the centralangle of the arc. The measurements of the main angle deserve to be provided in degrees or radians, and accordingly, we calculate the arc length of a circle.For acircle, the arc length formula isθ time theradius that a circle.

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The arc length formula in radians have the right to be express as, arc size = θ × r, whenθis in radian. ArcLength = θ × (π/180) × r, whereθis in degree, where,

L = length of anArcr = Radius the the circle

Arc length Formula in Radians

The arc length of a circle have the right to be calculate using different formulas, based upon the unit of the center angle of the arc. The arc length formula in radians can be express as,

ArcLength = θ × r

where,

L = Arc Lengthθ = center angle of the arc in radiansr = Radius of the circle

How to find Arc length of a Curve?

The arc size of an arc the a circle have the right to be calculation using different methods and formulas based on the offered data. Some important instances are provided below,


find arc length with the radius and main anglefind arc size without the radiusfind arc size without the central angle

How to discover Arc size With the Radius and central Angle?

The arc size of a circle can be calculated through the radius and central angle using the arc length formula,

Length of one Arc = θ × r, whereθis in radian.Length of an Arc= θ × (π/180) × r, whereθis in degree.

How to find Arc size Without the Radius?

The arc length of a circle have the right to be calculated there is no the radius using:

Central angle and the sector area:

Multiply the sector area through 2 and further, division the result by the central angle in radians.Find the square source of the result of the division.Multiply this obtained root through the central angle again to get the arc length.The systems of this calculated arc size will be the square source of the ar area units.

Example: calculation the arc length of a curve with sector area 25square units and the central angle as 2 radians.

We have,

Sector area = 25 units

Central edge =2 radians

Step 1:Sector area × 2 = 25× 2 = 50Step 2: 50/central edge = 50/2= 25Step 3:√25 = 5Step 4: 5× central angle =5× 2 = 10 units

Thus, arc size = 10 units

Central angle and the chord length:

Divide the central angle in radians by 2 and also further, perform the sine role on it.Divide the given chord size by twice the result of action 1. This calculation offers you the radius together result.Multiply the radius by the central angle to obtain the arc length.

Example: calculate the arc length of a curve, whose endpoints touch a chord the the one measuring 5 units. The central angle subtended by the arc is2 radians.

We have,

Chord length = 5 units

Central angle =2 radians

Step 1:Central angle/2 = 2/2 = 1Step 2: Sin(1) =0.841Step 3:Chord length/ (2× 0.841) = 5/ 1.682 = 2.973 systems = radiusStep 4: Arc length = radius × central angle = 2.973× 2 = 5.946 units

Thus, arc length = 5.946 units

How to discover Arc size Without the central Angle?

The arc size of a circle can be calculated there is no the angleusing:

Radius and the ar area:

Multiply the sector area by 2.Then division the result by the radius squared (the units should bethe same) to acquire the main angle in radians.Multiply the central angleby the radius to acquire the arc length.

Example: calculate the arc size of a curve v sector area 25square units and also radius together 2 units.

We have,

Sector area = 25 units

Central angle =2 units

Step 1:Sector area × 2 = 25× 2 = 50Step 2: 50/radius2 = 50/4= 12.5 = main angle(rad)Step 3: Arc length = radius× central angle = 2× 12.5= 25units

Thus, arc size = 25units

Radius and also chord length:

Divide the chord length by twicethe given radius.Find the station sine that the obtained result.Double the result of the train station sine to obtain the main angle in radians.Multiply the central angleby the radius to get the arc length.

Example: calculation the arc size of a curve, whose endpoints touch a chord the the circle measuring 5 units. The radius the the circleis2 units.

We have,

Chord length = 5 units

Central angle =2 units

Step 1:Chord length/(2× radius) = 5/(2× 2) = 1.25Step 2: Sin-1(1.25) =0.949Step 3:Central edge = 2× 0.949= 1.898radiansStep 4: Arc length = radius× central angle = 2× 1.898= 3.796units

Thus, arc length = 3.796units

Important Notes

Given listed below are key highlights top top the ide of arc length.

Arc size = θ × r, whereθis in radian.Arc length = θ × (π/180) × r, whereθis in degree.

Related subject on Arc Length:


Example 1: find the size of an arc cut off by a main angle the 4 radians in a circle v a radius that 6 inches.

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Solution:

Center angle, θ = 4radians, radius, r = 6 inch . Use the arc lengthformula,L = θ × r= 4 × 6= 24 inches. ∴ Arc length(PQ) = 24inches

Example 2:Find the length of one arc cut off by a main angle,θ = 40º in a circle v a radius the 4inches.

Solution:

Radius, r = 4inches , θ = 40º. Usage the arc lengthformula,L = π× (r) × (θ/180º)= π × (4) × (40º/180º)= 2.79inches. ∴ Arc length(P0) = 2.79inches