By Team Aakash Byju's | 16th November 2022
In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. Find (i) length of the arc (ii) area of the sector formed by arc (iii) area of the segment formed by the corresponding chord.
Given, the radius of the circle is 21 cm and the angle subtended by the arc at the centre is 60°.
We need to find the length of the arc XY, using the formula
Length of the arc
2πr
θ
360
o
=
2 ×
22
7
× 21 ×
60
360
o
=
= 22cm
22
7
o
..
.
The area of the sector formed by the arc XY can be calculated using the formula
Area of sector
=
θ
360
o
πr
2
= 231cm
2
=
×
22
7
× 21 × 21
60
360
o
o
Next, the area of the segment formed by chord XY can be found by subtracting the area of triangle CXY from the area of the sector formed by arc XY.
Area of S = Area of sector CXY - Area of ΔCXY
In ΔCXY, CX and CY are of the same length. Thus, the angles opposite these sides are the same.
i.e., ∠X = ∠Y = a (say) ∠X + ∠Y + ∠C = 180 (Sum of interior triangles) a + a + 60 = 180 => a = 60 ∴ ∠X = ∠Y = 60 = ∠C
o
o
o
o
o
Since all angles in ΔCXY are the same, it is an equilateral triangle.
Area of an equilateral triangle
=
√3
4
(side)
2
=
4
(21)
2
√3
=
cm
191
2
(Approx)
Area of S = Area of sector CXY - Area of ΔCXY
2
=> Area of S = 231 - 191 = 40 cm