Let r and h be the radius and height of the right circular cylinder with the open top.
So surface area of the cylinder S is given by,
S = πr2+2πrh
⇒ h =

S−πr2

2πr

... (i)
Let V be the volume, so
V = πr2h = πr2

(S−πr2)

2πr

= r

(S−πr2)

2

dV

dr

=

S

2

−

3πr2

2

... (ii)
for maxima or minima

dV

dr

= 0
⇒ S = 3πr2 or r = √

S

3π

Using this (i)
h =

2πr2

2πr

= r

d2V

dr2

= - 3 πr
= - 3π √

S

3π

< 0
So, r = √

S

3π

is a point of maxima
And in this case radius of base = height