Area of Hollow Cylinder with Examples and FAQs

Area of Hollow Cylinder

Before deriving the formula for a cylinder that is hollow, let us understand what a Hollow Cylinder is actually.

A hollow cylinder is a cylinder that is empty(hollow) from the inside. The cylinder has an external(outer) cylindrical layer, inside which lies an internal cylinder that encloses hollow space. These two layers are co-axial to each other. The cross-sectional space between the two cylinders forms the annular width of the cylinder.

Area

The area of a cylinder is defined as the total extent of the inner and outer surfaces of the cylinder and its bases. For a cylinder that is hollow, the space inside the cylinder is empty from inside and it also has a difference between the internal and external radii of the cylinder. The difference of the areas of external and internal cylinder gives the total area of a cylinder that is hollow.

The Surface Areas of  Hollow Cylinder

The surface area of a hollow cylinder can be of two types that are:

1. Curved(Lateral) Surface Area

2. Total Surface Area

Curved(Lateral) Surface Area

When a cylinder is hollow, the space inside the cylinder is empty(hollow) from inside and it also has a difference between the internal and external radii of the cylinder. Therefore, it will have two curved(lateral) surfaces one that is outside and the other inside.

Let r be the inner radius, R be the outer radius and H be the height of the hollow cylinder.

Let the circumference of the circular bases of the outer and inner surfaces of the cylinder be C and C’ respectively.

We know, the circumference of a circle = 2π(radius)

Hence, we have the circumference of the outer and inner surfaces as:

C = 2πR and C’ = 2πr

We know, Curved(Lateral) Surface Area = (Circumference of the base)×(Height of the object)

∴ Curved (Lateral) Surface Area of Outer Surface =  C × H = 2πRH

Similarly, the Curved(Lateral) Surface Area of the Inner Surface = C’ × H = 2πrH

Curved(Lateral) Surface Area 

= C.S.A. of Outer Surface + C.S.A. of Inner Surface

= 2πRH + 2πrH

= 2π(R + r)H

Total Surface Area(TSA)

The Total surface area of a hollow cylinder will be the sum of the curved(lateral) surface area and the area of the cross-sectional portion on the top and bottom faces of the cylinder. We have already got the formula for CSA of the hollow cylinder now let’s derive its cross-sectional area.

The Cross-sectional Area of a cylinder that is hollow, is the Area of the Ring formed on the top and bottom bases of the cylinder. The figure below represents what the top and bottom circular base looks like. 

Area of the outer circle =\pi R^{2}

Similarly, the area of the inner circle =\pi r^{2}

Hence the Area of cross-section(area of the circular ring) =\pi R^{2}-\pi r^{2}=\pi\left(R^{2}-r^{2}\right)

Since the area of the top and bottom, annular bases of the cylinder are equal, we have

∴ The total area of Cross-section(area of the top and bottom circular rings) 

=2 \times \pi\left(R^{2}-r^{2}\right)
=2 \pi\left(R^{2}-r^{2}\right)

As Total Surface Area(TSA) is the sum of the Curved Surface Area and Cross-sectional are we have:

Total Surface Area(TSA) 

= Lateral Surface Area + Area of Cross- Section

= [2 \pi(R+r) H]+\left[2 \pi\left(R^{2}-r^{2}\right)\right]

= 2 \pi\left[(R+r) H+\left(R^{2}-r^{2}\right)\right]
= 2 \pi[(R+r) H+(R+r)(R-r)]
= 2 \pi(R+r)[H+(R-r)]
= 2 \pi(R+r)(H+R-r)

Hence, the formula for Curved(Lateral) and Total Surface Area are:

Curved(Lateral) Surface Area of a Hollow Cylinder = 2 \pi(R+r) H

and

Total Surface Area of Hollow Cylinder = 2 \pi(R+r)(H+R-r)

Example

Determine the surface areas of a cylinder that is hollow, whose radii are 13 m and 15 m and its height is 12 m. Use pi = 22/7.

Solution

It is given that the inner radius of the cylinder, r = 13 m, the outer radius, R = 15 m and the height, H = 12 m.

∴ Curved Surface Area of the Hollow Cylinder

=2 \pi(R+r) H
=2 \times \frac{22}{7} \times(15+13) \times 12 \mathrm{~m}^{2}
=2 \times \frac{22}{7} \times 28 \times 12 \mathrm{~m}^{2}
=2 \times 22 \times 4 \times 12 \mathrm{~m}^{2}
=2112 \mathrm{~m}^{2}

Now calculating the Total Surface Area of the Cylinder

=2 \pi(R+r)(H+R-r)
=2 \times \frac{22}{7} \times(15+13) \times(12+15-13) \mathrm{~m}^{2}
=2 \times \frac{22}{7} \times 28 \times 14 \mathrm{~m}^{2}
=2 \times 22 \times 4 \times 14 \mathrm{~m}^{2}=2464 \mathrm{~m}^{2}

Therefore, the curved surface area of the hollow cylinder is 2112 \mathrm{~m}^{2}and its total surface area is 2464 \mathrm{~m}^{2}.