Area of Hollow Cylinder

A hollow cylinder, also known as a cylindrical shell, is a three-dimensional geometric shape characterized by two concentric cylinders, one inside the other. The area of a hollow cylinder includes the surface areas of both the inner and outer cylinders as well as the area of the circular ends. 

1.0What is a Hollow Cylinder?

A hollow cylinder is formed by removing a smaller, solid cylinder from a larger one, both sharing the same central axis. The remaining shape is the hollow cylinder, which can be thought of as a pipe or tube. Understanding the geometry of this shape is essential for calculating the area, which is used to determine various physical properties.

2.0Area of Hollow Cylinder Formula

To determine the area of a hollow cylinder, we primarily focus on two components:

Curved Surface Area (C.S.A) of the Hollow Cylinder

The curved surface area of a hollow cylinder is the sum of the curved surfaces of both the inner and outer cylinders. The formula to calculate the curved surface area is:

C.S.A = 2πRh + 2πrh = 2πh (R +r)

where:

• R is the outer radius,
• r is the inner radius,
• h is the height of the cylinder.

Total Surface Area (TSA) of the Hollow Cylinder

The total surface area includes the curved surface area and the areas of the top and bottom circular ends. The formula to calculate the total surface area is:

T. S. A = 2π (R + r) h + 2π (R² – r²)

= 2π (R + r) h +2π (R + r) (R – r)

= 2π (R + r) (R – r + h)

This formula accounts for both the curved surface and the cross-sectional area of the circular ends.

3.0Area of Hollow Cylinder with Diameter

When the diameter is provided instead of the radius, the formulas can be adjusted accordingly. The diameter D is related to the radius by D = 2R or R = \frac{D}{2}. Therefore, the formulas become:

• Curved Surface Area (CSA):

C.S.A = $2\pi h(\frac{D}{2}+\frac{d}{2})$

C.S.A = πh (D + d)

• Total Surface Area (TSA):

T.S.A =  $2\pi (\frac{D}{2}+\frac{d}{2})$ +  $2\pi ((\frac{D}{2}{)}^2-(\frac{d}{2}{)}^2)$

$=\pi (D+d)h+2\pi (\frac{D^2-d^2}{4})$

$=\pi (D+d)h+\frac{\pi }{2}(D-d)(D+d)$

$=\frac{\pi }{2}(D+d)(D-d+2h)$

where D and d are the diameters of the outer and inner cylinders, respectively.

4.0Cross Section Area of Hollow Cylinder

The cross-sectional area of a hollow cylinder is the area of the annular region between the outer and inner circles. It can be calculated using the formula:

Cross- Section Area = π(R² –r²)

This area is crucial when determining the flow capacity or the amount of material required to construct the hollow cylinder.

5.0Solved Examples on Area of Hollow Cylinder

Example 1: Calculate the curved surface area of a hollow cylinder has an outer radius of 7 cm, an inner radius of 5 cm, and a height of 10 cm. 

Solution:

Let the outer radius, the inner radius and the height be R, r and h respectively 

R= 7cm, r = 5 cm, h = 10 cm

Curved surface area of a hollow cylinder 

C.S.A = 2 πh (R + r)

= 2π × 10 × (7 + 5)

= 2π × 10 × 12 

= 240 π cm² 

So, the curved surface area of a hollow cylinder is 240 π cm².

Example 2: Calculate the total surface area of a hollow cylinder that has an outer radius of 8 cm. and inner radius of 5 cm, and a height of 12 cm.

Solution:

Let the outer radius, the inner radius and the height of the hollow cylinder by R, r and h respectively.

R = 8 cm, r = 5 cm, h = 12 cm

Total surface area of a hollow cylinder 

T.S.A = 2π (R + r) h + 2 π (R² – r²)

= 2 π (8 + 5) 12 + 2π (64 – 25)

= 2π (13) (12) + 2π (39)

= 312π + 78π = 390π cm² 

So, the total surface area of a hollow cylinder is 390 π cm².

Example 3: Find the cross-sectional area of a hollow cylinder with an outer diameter of 14 cm and an inner diameter of 10 cm. 

Solution:

Let the outer radius and the inner radius be R and r respectively

R = $\frac{14}{2}$ = 7 cm, r = $\frac{10}{2}$ = 5 cm

Cross sectional area of a hollow cylinder

= π (R² – r²) 

= π (49 – 25) = 24π cm² 

So, the Cross-sectional area of a hollow cylinder is 24π cm².

Example 4: A hollow cylinder is used as a pipe. The outer radius is 14 cm, the inner radius 12 cm and the length of the pipe is 5 meters. Calculate the curved surface area in square meters.

Solution:

Let the outer radius, inner radius and length of the pipe be R, r and h respectively.

R = 14 cm, r = 12 cm, h = 5 m = 500 cm

Curved surface area = 2πh (R +r)

C.S.A = 2π × 500 (14 + 12)

= 2π × 500 × 26

= 26000π cm² 

= 2.6π m² 

So, the curved surface area of pipe is 2.6π m²  

Example 5: Find the total surface area of a hollow cylinder with an outer diameter of 12 cm, an inner diameter of 8 cm, and a height of 20 cm.   

Solution:

Let the outer radius, the inner radius and height be R, r and h respectively.

R = 6 cm, r = 4 cm, h = 20 cm

Total surface area of a hollow cylinder

T.S.A = 2π (R + r) h + 2 π (R² – r²)

= 2 π (6 + 4) × 20 + 2 π (36 – 16)

= 400 π + 40 π 

= 440 cm²

So, the total surface area of a hollow cylinder is 440 cm².

6.0Practice Question on Area of Hollow Cylinder

1. A hollow cylinder has an outer diameter of 16 cm, an inner diameter of 12 cm, and a height of 20 cm. Find the curved surface area.
2. Determine the cross-sectional area of a hollow cylinder if the outer radius is 6 cm and the inner radius is 4 cm.
3. A hollow cylinder has an outer diameter of 18 cm, an inner diameter of 12 cm, and a height of 25 cm. Calculate the total surface area.
4. Find the curved surface area of a hollow cylinder with an outer radius of 9 cm, an inner radius of 6 cm, and a height of 30 cm.
5. A hollow cylinder has an outer radius of 5 cm and an inner radius of 3 cm. The height of the cylinder is 10 cm. What is the cross-sectional area?
6. Determine the total surface area of a hollow cylinder with an outer radius of 7 cm, an inner radius of 4 cm, and a height of 8 cm.

7.0Sample Questions on Area of Hollow Cylinder

1. What is the total surface area (TSA) of a hollow cylinder? 

Ans: The total surface area of a hollow cylinder includes both the curved surface area and the areas of the circular ends. The formula is:

T.S.A = 2π (R + r) h + 2π (R² – r²)

2. What is the cross-sectional area of a hollow cylinder?  

Ans: The cross-sectional area of a hollow cylinder is the area of the annular region between the outer and inner circles. It is calculated using the formula:

Cross- Section Area = π(R² –r²)