# 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 = ri = = 7 cm

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

external radius = re = 7 + 2 = 9 cm

Height of cylinder = h = 26 cm

Curved surface area of cylinder = 2πrh

external curved surface area of cylinder = 2πreh

inner curved surface area of cylinder = 2πrih

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 = πre2 – πri2

= π × (re2 – ri2)

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

ring area = π × (re+ ri) × (re– ri)

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

So total ring area = 2 × π × (re+ ri) × (re– ri)

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 = 2πreh + 2πrih + 2 × π × (re+ ri) × (re– ri)

Total surface area of hollow cylinder = 2πh(re + ri) + 2π(re+ ri)(re– ri)

Total surface area of hollow cylinder = 2π(re + ri)[h + re– ri]

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

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.