The area of a circle is 81π sq.cm. What is the length of the area subtending an angle of 300° at the centre and also the area of the corresponding sector?

 The formula for the area of a circle is given by A = πr^2, where A is the area of the circle and r is its radius. We are given that the area of the circle is 81π sq.cm, so we can solve for the radius:

A = πr^2

81π = πr^2

r^2 = 81

r = 9

So the radius of the circle is 9 cm.

To find the length of the arc subtending an angle of 300° at the center of the circle, we can use the formula L = rθ, where L is the length of the arc, r is the radius of the circle, and θ is the central angle in radians (since the formula uses radians, we need to convert 300° to radians).

To convert degrees to radians, we use the formula:

radians = (π/180) × degrees

So,

θ = (300/180) × π = 5π/3 radians

Therefore, the length of the arc is:

L = rθ = 9 × 5π/3 = 15π cm

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

Area of sector = (θ/2π) × πr^2

Substituting the values:

Area of sector = (5π/3 × 1/2π) × 9^2 = (5/6) × 81π = 45π sq.cm

Therefore, the length of the arc subtending an angle of 300° at the center of the circle is 15π cm, and the area of the corresponding sector is 45π sq.cm.