Question

A right circular cone has a base of R and slant height L. the curved surface area of this cone can be unfolded to form the sector of a circle. the radii of this sector subtend an angle of θ radians at its centre (i) find θ in terms of π, r and l (ii) hence show that the curved surface area of the cone is given by πrl

Answer 1

The angle θ in terms of π, R, and L is found by setting the arc length equal to the circumference of the base of the cone and then dividing by the slant height L; θ = 2πR / L. The curved surface area of the cone is πRL, demonstrated by using the formula for the area of a sector.

Step-by-step explanation:

To find the angle θ in terms of π, R, and L for the sector of a circle that represents the unfolded curved surface area of a right circular cone, we need to use the relationship between the arc length and the radius of a circle. For a full circle, the arc length is equal to the circumference, which is 2πR. Since the curved surface of the cone, when unfolded, is a sector of a circle, the arc length of the sector will be equal to the circumference of the base of the cone. The angle subtended by the radii of the sector, therefore, will be:

Δθ = θ = (Arc length) / (Radius) = (Circumference of cone's base) / (Slant height) = (2πR) / L

To show that the curved surface area of the cone is πRL, we use the formula for the area of a sector, which is (θ/2π) × (πL2). Plugging in our expression for θ, we get:

Curved Surface Area = ((2πR/L)/2π) × (πL2) = πRL