A solid cylinder has total surface area of 462 m². Its curved surface area is one–third of its total surface area. Find the volume of the cylinder.

Total surface area of cylinder = 462 m²

Curved surface area of cylinder = × Total surface area of cylinder

= × 462

= 154 m²

Curved surface area of cylinder = 154 m²

We know

Total surface area of cylinder = curved surface area + area of circles at top and bottom of cylinder

⇒ 462 = 154 + area of circles at top and bottom of cylinder

⇒ area of circles at top and bottom of cylinder = 462 – 154 = 308 m²

Let r be the radius of to and bottom circle of cylinder

area of circles at top and bottom of cylinder = 2πr²

⇒ 2πr² = 308

⇒

⇒ r² =

⇒ r² = 7 × 7

⇒ r = ± 7

r is radius and radius cannot be negative hence r = 7 m

now to find volume we need to find one more parameter about the cylinder which is the height

let us assume the height to be h

curved surface area of cylinder = 2πrh

⇒ 2πrh = 154

⇒ 2 × × 7 × h = 154

⇒ h =

⇒ h = m

volume of cylinder = πr²h

= × 7² ×

= 11 × 7 × 7

= 539 m³

Therefore, volume of cylinder is 539 m³