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Arc is a part of the circumference of a circle. Any smooth curve joining any two endpoints is called an arc. In simpler terms, an arc is one of the portions of a circle or a part of a curve. An arc can be a portion of any other curved figure like an ellipse but it mostly refers to a circle. In Mathematics, the length of an arc can be measured using two different ways. In case when the length of an arc is exactly half of the circle, it is referred to as a semicircular arc.
Read Also: Cyclic Quadrilateral
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Key Terms: Arc, Arc Length, Line Segment, Circumference, Radian, Degree, Radius, Curve, Chord, Minor Arc, Major Arc, Semicircular Arc
Meaning of Arc
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An arc is part of a circle that is less than the entire circle. Mostly we can see this arc in shapes like cylinders, circles, ellipse, etc. It is the part of Circumference of Circle.Any smooth curve joining any two points is called Arc.
Arc of a Circle
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Arc of a circle can be defined as a part of the circumference of the circle.A straight line that connects two ends points of the arc is known as a chord of a circle. The line that can be drawn from the centre point of the circle is called the radius. If the length of an arc is exactly half of the circle, it is known as a semi-circular arc.
Arc
Read More: Circles Important Questions
Symbol of Arc
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The arc is symbolized by ‘\(\cap\)’ or ‘\(\frown\)’. The arc in the above figure is called arc PQ or QP since the order of points is not important. These points can be written in the letter PQ with a curved line above it, and can be read as “Arc PQ”.
Types of Arc in a Circle
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There are different types of arcs in a circle:
- Minor Arc: An arc that measures less than 180° is called a minor arc.
- Semicircular Arc: An arc that measures exactly 180° is called a semicircular arc as it divides the circle into two parts.
- Major Arc: An arc that measures greater than 180° is known as Major Arc.
Read More: Areas Related to Circles
Measures of an Arc
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We can measure the arcs in two ways. They are as follows:
- The angle of the arc
- Length of the arc
The angle of the arc: The angle subtended by the arc at the centre of the circle is the angle of the arc. Using the arc length formula, we can solve or find the measure of arc angle.
Arc length = C(θ/360°)
θ = (Arc length/C)360°
Arc Length: Arc length is defined as the medium between two points on the circumference of the circle.
Arc Length
The length of the arc can be measured using the different formulas as per the angle of the arc. Central angle measurements are given in radians or degrees. In a circle, the arc length formula is said to be θ times the radius of a circle.
Read More: Arc Length Formula
If the central angle is measured using degrees, the arc length formula will be:
Arc length = 2πr(θ/360°)
Where θ indicates the central angle of the arc in degrees, r indicates the radius of the arc
As we know,
Circumference of the circle = 2πr
Therefore, length of the arc = C(θ/360°)
When the Angle is in Radians
When the central angle is given in radians, the arc length formula is given as
Arc Length = θ × r
Here, θ = Centre angle of the arc in radians, and r = Radius of the circle.
Read More: Area of Segment of a Circle
Things to Remember
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- Arc is a small curve joining two endpoints. It is a small part of the circumference of the circle.
- An arc is a portion of any other curved shape like an ellipse but it mostly refers to a circle.
- There are three types of arcs. They are major arcs, minor arcs and semi-circular arcs.
- The arc length formula is used to find the measure of distance along the curved line making up the arc which is arc length.
- The measurement of the central angle is usually given in radians or degrees.
- When the central angle is measured in degrees, the arc length formula is 2πr(θ/360)
- When the central angle is given in radians, the arc length formula is θ x r.
Sample Questions
Ques. The radius of the circle is given as 14 units and the arc subtends an angle of 65° at the centre. What will be the length of the arc? (3 Marks)
Ans. We know that,
Circumference of circle = 2πr
= 2π × 14 = 28π
Arc length = (θ/360) × C
= (65°/360°)28π
= 15.882 units
Ques. Write the formulas to find Arc Length with Radius and Central Angle? (3 Marks)
Ans. The arc length of a circle can be calculated using the given formulas:
- Length of an Arc = θ × r when the central angle is given in radians.
- Length of an Arc =2πr(θ/360), when the central angle is measured in degrees
Ques. Explain what do you understand by the arc length. (3 Marks)
Ans. Arc length is defined as the medium between two points on the circumference of any circle. It is a part of the circumference.
Subtended angle by an arc at any point is known to be the angle formed between the two-line segments joining that point to the endpoint of the arc.
Arc length is the distance that goes through the curved line of a circle. It is considered between two endpoints of a circle and a straight line. The length of an arc can be measured using different formulas depending on the unit of the central angle of the arc. Measurement of a central angle is usually given in radians or degrees.
Ques. Find the arc length of a circle that subtends an angle of 120° to the centre of a circle whose radius is 15cm. (3 Marks)
Ans. Here, radius=15cm
Angle =120°, π=3.14
The length of an arc = 2πr(θ/360)
= 2 x 3.14 x 15 x (120/360)
= 31.399 cm
Ques. Calculate what will be the area of the sector of a circle with a radius of 5 cm and of angle 60°. Also, Calculate the area of the corresponding major sector (Use π = 3.14) (3 Marks)
Ans. We know that,
Radius = r = 5 cm, θ=60°
Area of sector = [θ /360] × \(\pi r^2 \)
= (60/360) × 3.14 × (5)2
= (1/6) ×3.14× 5×5
= (1/6) ×3.14×25
= 78.5/6 cm2
= 13.083 cm2
Area of the major sector = [(360 – θ)/ 360] × πr2
= [(360 – 60)/360] × 3.14 × 5 × 5
= (300/360) × 3.14 × 25
= 65.41 cm2
Ques. The radius of the circle is 14 units and the arc subtends 60° at the centre. What is the length of the arc? (3 Marks)
Ans. We know that,
θ = 60° and radius r = 14
Circumference of circle = 2πr
C = 2π × 14 = 28π
arc length = (θ/360) × C = (60°/360°)28π = 14.653 units
Ques. Find the length of an arc cut off by a central angle of 4 radians in a circle with a radius of 6 inches. (3 Marks)
Ans. Centre angle, θ = 4 radians, radius r = 6 inches.
Use the arc length formula,
L = θ × r
= 4 × 6
= 24 inches.
∴ Arc length = 24 inches
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