An arc of a circle is a part of the circle’s circumference between two distinct points on it. It represents a curved portion of the boundary of the circle. When the endpoints of an arc are joined by a straight line, a chord is formed, and if the arc measures half of the circle, it is called a semicircle.
Construction of an Arc of a Circle
To make an arc, we can use the following steps:
- Pick any three distinct points that are not collinear.
- Draw straight lines connecting each pair of points.
- Construct the perpendicular bisectors of both lines and locate their point of intersection. This intersection point is the circumcircle's center.
- Choose the distance between the center and any of the three points as the radius.
- Using the chosen radius and the center point, draw a circle.
Types of Arcs
A circle is divided into two sections by an arc, as you must have observed.
1. Minor Arc
A circle's minor arc is essentially less than half of the circle's overall arc. The blue-colored curve in the following figure is the minor arc in the circle.
2. Major Arc
The major arc of the main circle is the arc that extends more than half of the circle. The red-colored curve in the following figure is the major arc in the circle.
Arc of the Circle Formula
The formula shown below can be used to determine an arc's length.
Arc Length of the Circle = 2πr(θ/360°)
where,
- r denotes the radius of the circle,
- 360° the angle of one full revolution, and
- θ which is the centre angle of the arc.
- π (Pi) has a value of 3.14.
Simplifying this formula further we get,
Arc Length of the Circle = (θ/360°) 2πr = (θ/180°) πr
Also Check:
Finding Length of Arc of a Circle
Assume we have a circle with a radius of 10 units and a center angle of 120°. The length of the arc subtended by this angle is what we're looking for.
Step 1: Check the Given values
- Radius (r) = 10 units
- Central angle (θ) = 120°
Step 2: Calculate Arc Length
Using the formula for arc length:
Arc Length = (θ/360°) × 2πrCalculate Fraction of Circle's Circumference
θ / 360° = 120°/360° = 1/3
Calculate Arc Length
Arc Length = (1/3) × 2π × 10
⇒ Arc Length = (2/3)π × 10
⇒ Arc Length = (20/3)π units
Step 3: Finalize the Result
If you want a numerical estimate, the value of π (pi) is about 3.14.
Thus, Arc Length ≈ (20/3) × 3.14 ≈ 20.93 units
The arc subtended by a central angle of 120° in a circle with a radius of 10 units is roughly 20.93 units long.
Finding the Arc Length in Radians
The angle that an arc occupies in radians and the proportion of the arc's length to the circle's radius are related. In this instance.
θ = (Length of an Arc)/(Radius of the Circle)
OR
S = r θ
where,
- θ is the angle in radians that an arc occupies,
- S is the angle's length, and
- r is the radius of the given circle.
- For θ = 1 radian, or s = r, it is the center angle that a radius-length arc subtends.
- The radian is merely another unit of measurement for angles. For instance, multiply the angle (in degrees) by π/180 to convert angles from degrees to radians.
- The angle (in radians) is multiplied by 180/π to convert from radians to degrees.
Angle Subtended by Arc at Center
The angle subtended by an arc at the center of a circle is the angular measure created by two radii commencing from the center and continuing to the arc's ends. It is the basic connection between the central angle and the appropriate arc length. This angle, given in radians or degrees, controls the length of the arc, with a direct ratio to the radius.
Theorem of Angle Subtended by Arc at Center
Given: An arc PQ of a circle is given, bounded by angles ∠POQ at the centre O and ∠PAQ at a point A on the remaining half of the circle.
To prove : ∠POQ=2∠PAQ
Construction : Join the line that AO extended to B.
Proof:
∠BOQ=∠OAQ+∠AQO . . . (1)
Also, in △ OAQ,
OA=OQ [The radius of a circle]
Therefore,
∠OAQ=∠OQA [Angles opposing equal sides have the same value]
∠BOQ=2∠OAQ . . . (2)
Similarly, BOP=2∠OAP . . . (3)
Adding 2 & 3, we get,
∠BOP+∠BOQ=2(∠OAP+∠OAQ)
∠POQ=2∠PAQ . . . (4)
For instance 3, where PQ is the main arc, equation 4 is substituted by Reflex angle, ∠POQ=2∠PAQ.
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Solved Examples
Example 1: Using 48 cm, determine the length of an arc of a circle that forms a 160° angle with the circle's center.
Given:
The value of π = 3.14
The value of θ = 160
The value of r = 48
The length of an arc = 2πr(θ/360)
= 2×3.14×48×160/360
= 133.97 cm.
Example 2: The radius of the circle is 18 units, and the arc subtends 85° at the center. How long is the arc, measured in terms of circumference?
Circumference C = 2πr = 2π × 18 = 36π
Arc length L = (θ/360) × C
= (85/360) × 36π
= (85/10)π
= 8.5π unitsNote: Since θ is in degrees, we used L = (θ/360) × 2πr.
(If θ were in radians, then L = rθ)
Example 3: Determine the length of an arc with a radius of 20 cm and an angle of 0.456 radians.
Given:
Radius (r) = 20
Radians (θ) = 0.456
Arc length = r θ
= 0.456×20
= 9.12 cm.
Example 4: A circle having a radius of 6 mm and a length of 15.06 mm should have an angle subtended by it.
Given:
Arc length = 15.06 mm
Radius(r) = 6 mm
Arc length = r θ
15.06 = 6 θ
Divide both sides by 6.
2.51 = θ
Thus, the arc there subtends an angle of 2.51 radians.
Practice Problems
Problem 1: Find the length of the arc of a circle with a central angle of 45° and a radius of 8 inches.
Problem 2: What is the length of the arc if the circle's radius is 6 centimeters and the arc's angle is 120°?
Problem 3: Given a circle with a radius of 10 meters, find the measure of the central angle if the length of the arc is 15 meters.
Problem 4: Calculate the length of an arc in a circle with a radius of 5 inches if the central angle is 60°.
Problem 5: A sector of a circle has a central angle of 90° and a radius of 12 centimeters. Find the length of the arc and the area of the sector.
Problem 6: Calculate the radius of a circle given that the arc length is 10 m and the central angle is 0.5 radians.
Problem 7: If an arc length is 5 cm and the radius of the circle is 7 cm, what is the central angle in degrees?
Problem 8: Determine the length of an arc with a radius of 20 cm and a central angle of 150 degrees.
Problem 9: Given a central angle of 2 radians and an arc length of 18 cm, find the radius of the circle.
Problem 10: Calculate the arc length of a circle with a radius of 10 cm and a central angle of 3 radians.