Question

Q. 19 The area of circle is \( 225 \pi sq . cm \). Find the length of its arc subtending an angle \( 120^{\circ} \) at the

Answer 1

Find the radius of the circle.

Area of circle = πr²

225 = πr²

r² = 75

r = √75 = 15 cm

Find the circumference of the circle.

Circumference of circle = 2πr

= 2π x 15

= 90 cm

Find the ratio of the arc's central angle to 360° and multiply this ratio by the circle's circumference to find the arc length.

Arc length = (θ/360°) x 2πr

= (120°/360°) * 90 cm

= 30 cm

Therefore, the length of the arc subtending an angle of 120° at the center of a circle with area 225 sq. cm is 30 cm.

To find the area of the corresponding sector, we can use the following formula:

Sector area = (θ/360°) x πr²

where θ is the sector's central angle and r is the circle's radius.

Substituting θ = 120° and r = 15 cm into the formula, 

we get:

Sector area = (120°/360°) * π * 15²

= (1/3) x π x 225

= 75π sq. cm

Therefore, the area of the corresponding sector is 75π sq. cm.