The area of a circle which subtends an angle of theta radians at the centre of the circle is 2/3m^2. If the length of the arc of the sector is 2m, Find theta

Answer 1

Let r be the radius of the circle. The area of the sector with angle θ is given by:

Area = (1/2) * r^2 * θ

According to the problem, this area is (2/3)m^2:

(1/2) * r^2 * θ = (2/3)m^2

The length of the arc is given by the formula:

Arc length = r * θ

According to the problem, this length is 2m:

r * θ = 2m

We have a system of two equations with two unknowns, r and θ. We can solve for one of the variables in terms of the other.

From the equation for the arc length, we have:

θ = 2m / r

Now we can substitute this into the equation for the area:

(1/2) * r^2 * (2m / r) = (2/3)m^2

Simplify the equation:

m * r = (2/3)m^2

Divide by m (assuming m ≠ 0):

r = (2/3)m

Now substitute this back into the equation for θ:

θ = 2m / ((2/3)m)

θ = 2m * (3/2m)

θ = 3 radians

So the angle θ is 3 radians.