In the previous section we completed the discussion on Area of circles. In this section we will see Length of Arcs.
■ Any part of a circle between two points on it, is called an arc.
• So AB is an arc. We denote it as 'arc AB'.
• But we notice that, just saying arc AB can cause confusion. For example, in fig.b, the green portion, as well as the yellow portion, can be called arc AB. To avoid such a confusion, we mark two more points ‘C’ and ‘D’ on the circle. Then we get two separate arcs: arc ADB and arc ACD.
• arc ACD is called the major arc and arc ABD is called the minor arc.
We want the length of the minor arc AB in cm. For that, we need the help of some angle measures.
In fig.21.30(a), the starting point A is joined to the centre O.
The end point B is also joined to the centre O.
■ Line joining a point on the circle or arc to the centre is called a radial line
• Let xo be the central angle of our arc AB. We can say that, when a point travels from A to B along the circumference of the circle, it ‘turns’ through an angle of xo. We are going to find the length of the arc AB with the help of this xo
Consider the circle in fig.21.29(a) below.
It has the centre at ‘O’, and a radius ‘r’ cm.
 |
| Fig.21.29 |
• A point
travels from point A to point B along the circumference of the
circle. We want to find the distance travelled by the point.
■ Any part of a circle between two points on it, is called an arc.
• So AB is an arc. We denote it as 'arc AB'.
• But we notice that, just saying arc AB can cause confusion. For example, in fig.b, the green portion, as well as the yellow portion, can be called arc AB. To avoid such a confusion, we mark two more points ‘C’ and ‘D’ on the circle. Then we get two separate arcs: arc ADB and arc ACD.
• arc ACD is called the major arc and arc ABD is called the minor arc.
We want the length of the minor arc AB in cm. For that, we need the help of some angle measures.
In fig.21.30(a), the starting point A is joined to the centre O.
 |
| Fig.21.30 |
■ Line joining a point on the circle or arc to the centre is called a radial line
■ The angle between the two
radial lines drawn through the end points of an arc is called the
central angle of the arc
■ This angle is also known as
the ‘angle subtended by the arc at the centre’
• Let xo be the central angle of our arc AB. We can say that, when a point travels from A to B along the circumference of the circle, it ‘turns’ through an angle of xo. We are going to find the length of the arc AB with the help of this xo
• Look at fig.21.29(b). The circle with
radius ‘r’ is divided into four equal parts using two green
lines. So P, Q, R and S are the quadrant points. Let us see four
cases:
Case 1:
1. When a point travels from P to
Q along the minor arc PQ, the distance covered is one fourth of the
total perimeter of the ‘circle with radius r cm’.
2. We know that the total
perimeter = 2πr. So the distance covered = (1⁄4 × 2πr)
3. Also, when the above distance is covered, we can say, the point ‘turns’ through an angle of 90o
4. So we can write: 90o → (1⁄4 × 2πr)
5. So 1o → [(1⁄4 × 2πr) ÷ 90]
6. [(1⁄4 × 2πr) ÷ 90] = 2πr⁄(4×90) = πr⁄180
7. So 1o → πr⁄180. That means, when the point turns through 1o, the distance covered along the circumference is πr⁄180 cm.
8. Another form of writing this is: If an arc of a circle of radius r, subtends an angle of 1o at the centre, the length of that arc is πr⁄180 cm.
Case 2:
8. Another form of writing this is: If an arc of a circle of radius r, subtends an angle of 1o at the centre, the length of that arc is πr⁄180 cm.
Case 2:
1. When a point travels from P to R along the arc PQR, the distance covered is one half of the total perimeter of the ‘circle with radius r cm’.
2. We know that the total perimeter = 2πr. So the distance covered = (1⁄2 × 2πr)
3. Also, when the above distance is covered, we can say, the point ‘turns’ through an angle of (90+90) = 180o
4. So we can write: 180o → (1⁄2 × 2πr)
5. So 1o → [(1⁄2 × 2πr) ÷ 180]
6. [(1⁄2 × 2πr) ÷ 180] = 2πr⁄(2×180) = πr⁄180
7. So 1o → πr⁄180. That means, when the point turns through 1o, the distance covered along the circumference is πr⁄180 cm
8. Another form of writing this is: If an arc of a circle of radius r, subtends an angle of 1o at the centre, the length of that arc is πr⁄180 cm.
Case 3:
7. So 1o → πr⁄180. That means, when the point turns through 1o, the distance covered along the circumference is πr⁄180 cm
8. Another form of writing this is: If an arc of a circle of radius r, subtends an angle of 1o at the centre, the length of that arc is πr⁄180 cm.
Case 3:
1. When a point travels from P to S along the arc PQRS, the distance covered is three fourth of the total perimeter of the ‘circle with radius r cm’.
2. We know that the total perimeter = 2πr. So the distance covered = (3⁄4 × 2πr)
3. Also, when the above distance is covered, we can say, the point ‘turns’ through an angle of (90+90+90) = 270o
4. So we can write: 270o → (3⁄4 × 2πr)
5. So 1o → [(3⁄4 × 2πr) ÷ 270]
6. [(3⁄4 × 2πr) ÷ 270] = 6πr⁄(4×270) = πr⁄180
7. So 1o → πr⁄180. That means, when the point turns through 1o, the distance covered along the circumference is πr⁄180 cm
8. Another form of writing this is: If an arc of a circle of radius r, subtends an angle of 1o at the centre, the length of that arc is πr⁄180 cm.
Case 4:
7. So 1o → πr⁄180. That means, when the point turns through 1o, the distance covered along the circumference is πr⁄180 cm
8. Another form of writing this is: If an arc of a circle of radius r, subtends an angle of 1o at the centre, the length of that arc is πr⁄180 cm.
Case 4:
1. When a point travels from P, and returns back to P along the arc PQRSP, the distance covered is one full of the total perimeter of the ‘circle with radius r cm’.
2. We know that the total perimeter = 2πr. So the distance covered = (2πr)
3. Also, when the above distance is covered, we can say, the point ‘turns’ through an angle of (90+90+90+90) = 360o
4. So we can write: 360o → (2πr)
5. So 1o → [(2πr) ÷ 360]
6. [(2πr) ÷ 360] = 2πr⁄(360) = πr⁄180
7. So 1o → πr⁄180. That means, when the point turns through 1o, the distance covered along the circumference is πr⁄180 cm
8. Another form of writing this is: If an arc of a circle of radius r, subtends an angle of 1o at the centre, the length of that arc is πr⁄180 cm.
■ In all the four cases, we get the same result. We can write it in the form of a theorem:
Theorem 21.1:
• A point is situated on the circumference of a circle with radius r
• It turns through an angle of 1o about the centre of the circle
• Then the distance travelled by it along the circumference of the circle is πr⁄180 cm
• Another form of writing this is: If an arc of a circle of radius r, subtends an angle of 1o at the centre, the length of that arc is πr⁄180 cm.
• So, if an arc in a circle of radius r, subtends an angle of xo at the centre, the length of that arc
= (πr⁄180 × x) = πrx⁄180 cm.
Based on the above theorem, we can write:
In fig.21.30(a), the length of the arc AB = πrx⁄180 cm
A sample calculation:
In a circle of radius 3 cm, an arc subtends an angle of 60o at the centre. What is the length of the arc?
Solution:
1. According to theorem 21.1 above, for every 1o turn, the length of arc on a circle of radius r will be equal to πr⁄180 cm..
2. So for 60, the length of arc on a circle of radius 3 cm = (3π×60)⁄180 cm = π cm = 3.14 cm
7. So 1o → πr⁄180. That means, when the point turns through 1o, the distance covered along the circumference is πr⁄180 cm
8. Another form of writing this is: If an arc of a circle of radius r, subtends an angle of 1o at the centre, the length of that arc is πr⁄180 cm.
■ In all the four cases, we get the same result. We can write it in the form of a theorem:
Theorem 21.1:
• A point is situated on the circumference of a circle with radius r
• It turns through an angle of 1o about the centre of the circle
• Then the distance travelled by it along the circumference of the circle is πr⁄180 cm
• Another form of writing this is: If an arc of a circle of radius r, subtends an angle of 1o at the centre, the length of that arc is πr⁄180 cm.
• So, if an arc in a circle of radius r, subtends an angle of xo at the centre, the length of that arc
= (πr⁄180 × x) = πrx⁄180 cm.
Based on the above theorem, we can write:
In fig.21.30(a), the length of the arc AB = πrx⁄180 cm
A sample calculation:
In a circle of radius 3 cm, an arc subtends an angle of 60o at the centre. What is the length of the arc?
Solution:
1. According to theorem 21.1 above, for every 1o turn, the length of arc on a circle of radius r will be equal to πr⁄180 cm..
2. So for 60, the length of arc on a circle of radius 3 cm = (3π×60)⁄180 cm = π cm = 3.14 cm
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