A hollow cylinder is a 3D shape that’s empty inside with some thickness around the edges. The formula for its total surface area (TSA) is:
TSA of Hollow Cylinder = 2πh (R + r) + 2π(R² – r²)
Here are formulas for other areas of a hollow cylinder:
- Lateral Surface Area (LSA): 2πh (R + r)
- Cross-Sectional Area: π(R2 − r2)
where,
- R is Radius of Outer Circle
- r is Radius of Inner Circle
What is Hollow Cylinder?
A hollow cylinder is a cylinder that is hollow from the inside. A hollow cylinder is defined as a three-dimensional object that is empty from the inside. In a hollow cylinder, there are two circular bases in the shape of rings. The circular base has two radii a smaller inner radius and a bigger outer radius.
Elements of a Hollow Cylinder
Hollow Cylinder Definition
A hollow cylinder is defined as a cylinder that is empty from the inside and has a difference between the internal and external radius.
There is some thickness enclosed between the inner radius and the outer radius, of the hollow cylinder, the thickness between them is equal to the difference between the internal and external radius. The height, of the hollow cylinder, is the perpendicular distance between its two circular bases.
Some examples of hollow cylinders are water pipes, straws, water bottles, etc.
Finding area of a hollow cylinder is similar to finding area of a cylinder. A hollow cylinder has two types of surface areas, i.e., a curved surface area and a total surface area.
- Curved surface area, or lateral surface area, is the surface area of the curved surface of the hollow cylinder.
- Total surface area of the hollow cylinder is the sum of its curved surface area and the areas of its two circular bases.
Curved Surface Area of Hollow Cylinder
Now, let’s calculate the curved surface area of the hollow cylinder. The curved surface area (CSA) of the hollow cylinder is equal to the sum of the external surface area (ESA) and the internal surface area (ISA) of the cylinder.
Let C1 be the outer circumference and C2 be the inner circumference of the given cylinder.
Thickness of the hollow cylinder (t) = R − r
We know that,
Circumference of a circle (C) = 2πr
So, C1 = 2πR and C2 = 2πr
CSA = ESA + ISA
We know that,
Curved Surface Area of a solid cylinder = C × h = 2πrh
CSA = 2πR × h + 2πr × h
CSA= 2πRh + 2πrh
CSA = 2πh (R + r)
Curved Surface Area of Hollow Cylinder = 2πh (R + r) square units
where,
- “h” is Height of Hollow Cylinder
- “R” is Outer Radius of Cylinder
- “r” is Inner Radius of Cylinder
Total Surface Area of Hollow Cylinder
Total surface area A of a hollow cylinder is the sum of the areas of its outer curved surface and its inner curved surface, along with the areas of the two circular bases.
Total Surface Area = Curved Surface Area + Areas of Bases
Let A1 be the area enclosed by the inner radius, r, and A2 be the area enclosed by the outer radius, R.
We know that the area of a circle (A) = πr2
So, A1 = πR2 and A2 = πr2
Now, the cross-sectional area of the base of the hollow cylinder (A) = A1 − A2
A = πR2 − πr2 = π((R² – r²)
TSA = CSA + 2 × A
= 2πh (R + r) + 2 × π(R² – r²)
TSA = 2πh (R + r) + 2π(R² – r²)
Total Surface Area of Hollow Cylinder = [2πh (R + r) + 2π(R² – r²)] square units
where,
- “h” is Height of Hollow Cylinder
- “R” is Outer Radius of Cylinder
- “r” is Inner Radius of Cylinder
Total Surface Area of Hollow Cylinder with One Side Open
A hollow cylinder with one side open is essentially a tube-like structure with an open end. This type of cylinder is commonly found in various industrial, architectural, and household applications, such as bottles and containers.
The total surface area of a hollow cylinder with one side open includes the outer curved surface area, the inner curved surface area, and the area of the closed end. To calculate the total surface area, we consider the following components:
- Outer Curved Surface Area (OCSA): This is the surface area of the outer curved surface of the cylinder. It is calculated using the formula 2πRh, where R is the outer radius of the cylinder and h is the height.
- Inner Curved Surface Area (ICSA): Similarly, the inner curved surface area of the cylinder is calculated using the same formula 2πrh, where r is the inner radius of the cylinder.
- Area of Closed End (ACE): Since one end of the cylinder is closed, we need to consider the surface area of the circular closed end. The area of the closed end is calculated using the formula 2πr2, where r is the inner radius.
Therefore, the total surface area (TSA) of the hollow cylinder with one side open is the sum of the outer curved surface area, the inner curved surface area, and the area of the closed end:
TSA = OCSA + ICSA + ACE
TSA = 2πRh + 2πrh + πr2
Total Surface Area of Hollow Cylinder with No Sides Open
When considering a hollow cylinder with no sides open, we are essentially referring to a solid cylinder with two closed ends. In this case, the total surface area includes the outer curved surface, the inner curved surface, and the areas of both the top and bottom ends.
To calculate the total surface area of a hollow cylinder with no sides open, we need to consider the following components:
- Outer Curved Surface Area (OCSA): The outer curved surface area of the cylinder is given by 2πRh, where R is the outer radius of the cylinder and h is the height.
- Inner Curved Surface Area (ICSA): The inner curved surface area of the cylinder is also 2πrh, where r is the inner radius.
- Area of Top and Bottom Ends: Since both ends of the cylinder are closed, we need to account for the areas of both the top and bottom circular ends. Each end has an area of 2πR2.
Therefore, the total surface area (TSA) of the hollow cylinder with no sides open is the sum of the outer curved surface area, the inner curved surface area, and the areas of both top and bottom ends:
TSA = OCSA + ICSA + Area of Top End + Area of Bottom End
TSA = 2πRh + 2πrh + πR2 + πR2
TSA = 2πRh + 2πrh + 2πR2
How to Find Area of Hollow Cylinder?
Let’s take an example to understand how to calculate the area of a hollow cylinder.
Example: Calculate the area of a hollow cylinder whose external radius is 12 cm, the internal radius is 9 cm, and the height is 7 cm.
Solution:
Step 1: Note the values of the given dimensions. Here, the external radius (R) is 12 cm, the internal radius (r) is 9 cm, and the height (h) is 7 cm.
Step 2: We know that the formula to find the area (TSA) of a hollow cylinder is [2πh (R + r) + 2π(R² – r²)] square units. Now, substitute the given values in the formula.
Step 3: Thus, the area of a hollow cylinder is calculated as
TSA = 1319.469 Square Meters
Questions on Area of Hollow Cylinder
Question 1: Calculate the area of a hollow cylinder whose external radius is 8 cm, the internal radius is 4 cm, and the height is 12 cm. [Use π= 22/7]
Solution:
- External Radius = 8 cm
- Internal Radius = 4 cm
- Height = 12 cm
We know that,
Area of Hollow Cylinder = 2πh (R + r) + 2π(R2 – r2)
= 2 × (22/7) × 12 × (8 + 4) + 2 × (22/7) × (82 – 42)
= 2 × (22/7) × 10 × 15 + 2 × (22/7) × 48
= 905.142 + 301.714
= 1,206.856 sq. cm.
Hence, area of a hollow cylinder is 1,206.856 sq. cm.
Question 2: Calculate the area of a hollow cylinder whose external radius is 6 m, the internal radius is 2 m, and the height is 8 m [Use π= 22/7]
Solution:
- External Radius = 6 m
- Internal Radius = 2 m
- Height = 8 m
We know that,
Area of the hollow cylinder = 2πh(R + r) + 2π(R2 – r2)
= 2 × (22/7) × 8 × (6 + 2) + 2 × (22/7) × (62 – 22)
= 2 × (22/7) × 8 × 8 + 2 × (22/7) × (36-4)
= 2 × (22/7) × 64 +2 × (22/7) × 32
= 352/7 × 64 + 64/7 × 32
=22528/7 + 2048/7
= 24576/7
= 3503.429 sq .m
Hence, area of a hollow cylinder is 603.429 sq. m.
Question 3: Ram has a hollow cylindrical pipe with him, and he was asked to find its curved surface area. The external radius of the pipe is 10 inches, the internal radius is 6 inches, and the height is 14 inches. [Use π= 22/7]
Solution:
- External Radius = 6 m
- Internal Radius = 2 m
- Height = 8 m
We know that,
Curved Surface Area of a Hollow cylinder = 2πh (R + r)
= 2 × (22/7) × 14 × (10 + 6)
= 2 × (22/7) × 14 × 16
= 1,408 sq. in
Hence, curved surface area of given hollow cylindrical pipe is 1,408 sq. in.
Question 4: Calculate the curved surface area of a hollow cylinder whose outer diameter is 26 cm, the inner diameter is 18 cm, and the height is 12 cm. [Use π= 22/7]
Solution:
Outer Diameter = 26 cm
- So, external Radius (R) = 26 cm/ 2 = 13 cm
Inner Diameter = 18 cm
- So, Internal Radius (r) = 18 cm/2 = 9 cm
Height = 12 cm
We know that,
Curved Surface Area of a Hollow cylinder = 2πh (R + r)
= 2 × (22/7) × 12 × (13 + 9)
= 2 × (22/7) × 12 × 22
= 1,659.428 sq. cm
Hence, curved surface area of given hollow cylindrical pipe is 1,659.428 sq. cm.
Examples of Hollow Cylinder
Following are a few examples of hollow cylinders:
- Pipes and Tubes: Pipes used in plumbing systems and tubes used in various industrial applications are examples of hollow cylinders. They have a cylindrical shape with a hollow interior, allowing fluids, gases, or other materials to pass through.
- Paper Towel Rolls: Paper towel rolls typically have a hollow cylindrical shape. The cardboard tube in the center provides structural support and allows the paper towels to be rolled onto and dispensed from the tube.
- Oil Barrels: Standard oil barrels used for storage and transportation of oil and other liquids have a hollow cylindrical shape. They are made of metal and have a sealed top and bottom to prevent leakage.
- Hollow Drums: Musical instruments like certain types of drums, such as the steelpan or hang drum, have a hollow cylindrical shape. The sound is produced by striking or vibrating the surface of the drum.
- Roller Bearings: Roller bearings used in machinery and automotive applications often have a hollow cylindrical shape. The inner and outer rings of the bearing form hollow cylinders, and the rolling elements fit between them.
FAQs on Hollow Cylinder
What is a hollow cylinder? Give some examples of a hollow cylinder.
A hollow cylinder can be defined as a three-dimensional geometric object that is empty from the inside. A hollow cylinder consists of two circular bases that have inner and outer radii. Straws, water pipes, tubes, toilet paper rolls, etc. are some examples of hollow cylinders that we see in our daily lives.
What is the formula to calculate the thickness of a hollow cylinder?
Thickness of a hollow cylinder is the enclosed space between the inner radius and the outer radius, which is equal to the difference between the internal and external radius.
Thickness of hollow cylinder (t) = R − r
What is formula to calculate curved surface area of a hollow cylinder?
Curved surface area, or lateral surface area, is the surface area of the curved surface of a hollow cylinder that is equal to the sum of the external surface area (ESA) and the internal surface area (ISA) of the cylinder.
Curved Surface Area of Hollow Cylinder = 2πh (R + r) square units
What is annular ring of a hollow cylinder?
Circular bases of a hollow cylinder are similar to an annular ring, which is a region bounded by two concentric circles. So, area of the circular base of the hollow cylinder is equal to the area of the annular ring of the cylinder.
What is formula to calculate volume of a hollow cylinder?
Formula to calculate the volume of a hollow cylinder is given as follows:
Volume of Hollow Cylinder = π(R2 – r2)h cubic units
What is area of hollow cylinder face?
Area of the two faces of a hollow cylinder is the difference between the outer and inner circular faces.
Area of Two Faces = 2π(r2outer − r2inner)
What is area of a half cylinder?
Area of a half cylinder includes curved surface area and the area of the circular base. It can be calculated using the formula (1/2)×2πr2 + πrh.
What is cylindrical area of cylinder?
Cylindrical area of a cylinder refers to the total surface area of the cylinder, including both the curved surface area and the area of the two circular bases. It can be calculated using the formula 2πr2+2πrh
What is the cross sectional area of a cylinder?
Coss-sectional area of a cylinder is the area of the circular shape formed by slicing the cylinder perpendicular to its axis. It is equal to the area of the base circle and can be calculated using the formula 2πr2
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