Write whether True or False and justify your answer

If the radius of a cylinder is doubled and its curved surface area is not changed, the height must be halved.

Let radius of the cylinder = r

Height of the cylinder = h

Then, curved surface area of the cylinder, CSA = 2πrh

According to the question, radius is doubled and curved surface area is not changed. We can write as,

New radius of the cylinder (say, R) = 2r

& new curved surface area of the cylinder, CSA’ = 2πrh [i.e., unchanged] …(i)

Alternate case: when R = 2r and CSA’ = 2πrh

But curved surface area of cylinder in this case, CSA’= 2πRh = 2π(2r)h = 4πrh …(ii)

Comparing equations (i) and (ii), we can say

equation (i) ≠ equation (ii)

as 2πrh ≠ 4πrh

Thus, if h = h/2 (height is halved) then,

CSA’ = 2π(2r)(h/2) = 2πrh

It matched CSA’ in equation (i).

Hence, it is true.