Prove that for a cone made by rolling up a semicircle, the area of the curved surface is twice the base area.

Let the radius of the sector be r_s

Central angle θ is 180° (∵ the sector is a semicircle)

Let the radius of the base be r_b

So circumference of the base = 2πr_b …(1)

This circumference is equal to the length of the arc

Since, the arc length of a semicircle is 'half the circumference of the

full circle'

The 'circumference of the full circle' is 2πrs.

So half of it is πrs. …(2)

Equating the results in (1) and (2):

2πr_b = πr_s

⇒ 2r_b = r_s. ..(3)

Find the area of the sector

The area of the sector is area of the semicircle which is

Let us substitute for r_s using the result in (3). We get:

= 2πr_b²

Area of the sector is same as the area of curved surface.

Area of the curved surface of the cone = 2πr_b² …(4)

To find the base area.

radius of the base of the cone = r_b.

So area of the base of the cone = πr_b². …(5)

Comparing the results in (4) and (5), we get:

Area of the curved surface of the cone = Twice the base area