# In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. Find:

(i) the length of the arc

(ii) area of the sector formed by the arc

(iii) area of the segment formed by the corresponding chord

**Solution:**

In the circle with radius r and the angle at the centre with degree measure of θ:

(i) Length of the Arc = θ/360° × 2πr

(ii) Area of the sector = θ/360° × πr^{2}

(iii) Area of the segment = Area of the sector - Area of the corresponding triangle

Draw a figure to visualize the problem

Here, r = 21 cm, θ = 60°

Visually it’s clear from the figure that;

Area of the segment = Area of sector AOPB - Area of ΔAOB

Radius, r = 21 cm, θ = 60°

(i) Length of the Arc, APB = θ/360° × 2πr

= 60°/360° × 2 × 22/7 × 21 cm

= 22 cm

(ii) Area of the sector, AOBP = θ/360° x πr^{2}

= 60°/360° × 22/7 × 21 × 21 cm^{2}

= 231 cm^{2}

(iii) Area of the segment = Area of the sector AOBP - Area of the triangle AOB

To find the area of the segment, we need to find the area of ΔAOB

In ΔAOB, draw OM ⊥ AB.

Consider ΔOAM and ΔOMB,

OA = OB (radii of the circle)

OM = OM (common)

∠OMA = ∠OMB = 90° (Since OM ⊥ AB)

Therefore, ΔOMB ≅ ΔOMA (By RHS Congruency)

So, AM = MB (Corresponding parts of the congruent triangles are always equal)

∠OMB = ∠OMA = 1/2 × 60° = 30°

In ΔAOM,

cos 30° = OM/OA or sin 30° = AM/OA

√3/2 - OM/r - or -1/2 = AM/r

OM = √3/2 r or AM = 1/2 r

AB = 2AM

AB = 2 × 1/2 r

AB = r

Therefore, area of ΔAOB = 1/2 × AB × OM

= 1/2 × r × √3/2 r

= 1/2 × 21 cm × √3/2 × 21 cm

= 441√3/4 cm^{2}

Area of the segment formed by chords = (231 - 441√3/4) cm^{2}

**Video Solution:**

## In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. Find: (i) the length of the arc (ii) area of the sector formed by the arc (iii) area of the segment formed by the corresponding chord

### NCERT Solutions Class 10 Maths - Chapter 12 Exercise 12.2 Question 5:

In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. Find: (i) the length of the arc (ii) area of the sector formed by the arc (iii) area of the segment formed by the corresponding chord.

The length of the arc APB, area of the sector AOBP and area of the segment of a circle of radius 21 cm in which an arc subtends an angle of 60° at the centre are 22 cm, 231 cm^{2} and (231-441√3/4) cm^{2} respectively.