A hollow cylinder, open at both ends has a thickness of 2 cm. If the inner diameter is 14 cm and its height is 26 cm then calculate the total surface of the hollow cylinder.

Cross section of cylinder i.e. top/bottom of cylinder will look like as shown

Inner diameter of cylinder = 14 cm

⇒ Inner radius = r_i = = 7 cm

As the thickness is 2 cm it can be seen from figure that external radius = inner radius + 2

⇒ external radius = r_e = 7 + 2 = 9 cm

Height of cylinder = h = 26 cm

Curved surface area of cylinder = 2πrh

⇒ external curved surface area of cylinder = 2πr_eh

⇒ inner curved surface area of cylinder = 2πr_ih

As seen the figure the area between the external circle and inner circle will also be counted in total surface area

That area is given by subtracting inner circle area from external circle area

Let us call that area as ring area

⇒ ring area = πr_e² – πr_i²

= π × (r_e² – r_i²)

Using identity a² – b² = (a + b)(a – b)

⇒ ring area = π × (r_e+ r_i) × (r_e– r_i)

We have two such ring areas at the top and bottom of cylinder

So total ring area = 2 × π × (r_e+ r_i) × (r_e– r_i)

Now,

Total surface area of hollow cylinder = external curved surface area of cylinder + inner curved surface area of cylinder + total ring area

⇒ Total surface area of hollow cylinder = 2πr_eh + 2πr_ih + 2 × π × (r_e+ r_i) × (r_e– r_i)

⇒ Total surface area of hollow cylinder = 2πh(r_e + r_i) + 2π(r_e+ r_i)(r_e– r_i)

⇒ Total surface area of hollow cylinder = 2π(r_e + r_i)[h + r_e– r_i]

Substituting values

⇒ Total surface area of hollow cylinder = 2 × × (9 + 7)[26 + 9– 7]

= 2 × × 16 × 28

= 32 × 22 × 4

= 2816

Therefore, total surface of the hollow cylinder is 2816 cm²