A cylinder is a three-dimensional geometric figure. It is considered a prism having a circle as its base. There are 2 types of cylinder named solid cylinder and hollow cylinder. A hollow cylinder is a figure that looks similar to a cylinder but is hollow from inside. You may have seen a water pipe which is a good example of a hollow cylinder. It has all the properties of a solid cylinder and some more. We often need to find the area of hollow cylinder given certain parameters.

A hollow cylinder can be defined as a three-dimensional figure which is hollow on the inside and has a radius difference between the interior and outside radius. It is a type of cylinder that is considered a form of a cylinder that is vacant on the inside. A hollow pipe is the best example of a hollow cylinder where the solid shape contains an inner and an outer radius.

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A hollow cylinder has a base that is similar to a circular ring. The area of cross-section is constant throughout the height of the cylinder therefore, there are two bases, one at the bottom of the cylinder and another at the top.

The upper and lower bases are circular. Although it looks like a solid cylinder from outside but it is hollow from inside.

Area of hollow cylinder is the amount of surface that the sides of the cylinder cover in two-dimensional space. It is a 2D property for any 2D or 3D geometric figure. There are different types of surface area depending on which part of the hollow cylinder we consider.

Inner and Outer Surface Area of Hollow Cylinder

Inner and outer surface area of hollow cylinder is the amount of area covered by the inner or outer curved surface of the hollow cylinder. Here we do not consider the area of the bases but only the lateral surface.

In the above image, we see a hollow cylinder. The red-colored portion indicates the inner surface and the black-colored portion indicates the outer surface of the hollow cylinder.

In the above image, a hollow cylinder is given.

So find the inner surface area of this cylinder we consider the inner radius which is r and the height h of the cylinder. Here we consider only the inner curved portion of the cylinder and not the outer one.

Therefore we get, A=2πrh

Similar to this, the outer surface area is the area covered by the outer portion of the cylinder only and we consider the outer radius R along with the height h this time.

Therefore we get, A=2πRh

Here A represents the surface area for that particular portion.

Curved Surface Area of Hollow Cylinder (Lateral Surface Area of Hollow Cylinder)

Curved surface area of hollow cylinder is the sum of the inner surface area and the outer surface area. It is also represented as CSA. It is also called as Lateral Surface Area of hollow cylinder as only lateral surfaces are considered.

In the above image, we consider the same hollow cylinder with an inner radius r and outer radius as R and height as h.

Thus according to the definition, the curved surface area of hollow cylinder or the lateral surface area of hollow cylinder will beCSA = 2πrh + 2πRh = 2π(r + R)h

We use the above formula to compute the curved surface area of hollow cylinder.

Cross-Sectional Area of Hollow Cylinder

If we cut any figure and see from the top in a 2D form, then the shape we get is known as the cross-section of that figure. Suppose you have a water pipe and then you cut it from the middle and see from the top. You will then see a ring shape which is the cross-section of that pipe.

Similarly, the cross-section of a hollow cylinder is a ring that we obtain after cutting it.

In the above images, we can see the cross-section of a hollow cylinder which is a ring. Now we need to find the area of this ring.

In the above image, we have taken the cross-section of a hollow cylinder and we will find the area of the highlighted portion. So we simply find the area of the inner circle and subtract it from the area of the outer circle to get the area of the highlighted path.

Therefore cross-sectional area of a hollow cylinder, A = πr₂²h − πr₁²h = π(r₂² − r₁²)

Here A represents the cross-sectional area of the cylinder.

Total Surface Area of Hollow Cylinder

The total surface area of hollow cylinder is the sum of all types of areas possible for a hollow cylinder. In other words, if we add the curved surface area with the area of the two bases, then we get the total surface area of a hollow cylinder.

Here as we know that the bases are nothing but a ring whose area we can find through the cross-section formula. But you should note that we need to add this cross-sectional area twice because there are two bases for a hollow cylinder.

So, if we again consider the same cylinder as shown above, then the total surface area of the hollow cylinder will be TSA=CSA+2 x cross-sectional area.

Therefore, \( TSA=2\pi\left(R+r\right)h+2\pi\left(R^2-r^2\right) \)

1. Hollow Cylinder with One Side Open

This type of cylinder is open at one end and closed at the other end. You’ll find these in things like water bottles or some containers.

To find the total surface area, you need to add:

• Outer Curved Surface Area: The area around the outside
  Formula: 2πRh (R = outer radius, h = height)
   
• Inner Curved Surface Area: The area on the inside
  Formula: 2πrh (r = inner radius, h = height)
   
• Area of the Closed End (the bottom circle)
  Formula: πr² (r = inner radius)
   

So, Total Surface Area (TSA) = 2πRh + 2πrh + πr²

2. Hollow Cylinder with No Sides Open

This cylinder is closed at both ends, like a sealed tube or tank.

To find the total surface area, add:

• Outer Curved Surface Area: 2πRh
   
• Inner Curved Surface Area: 2πrh
   
• Top and Bottom Ends: Each end is a circle with area πR², so together it's 2πR²

So, Total Surface Area (TSA) = 2πRh + 2πrh + 2πR²

Area of Hollow Cylinder Formula

The formula for different types of areas for a hollow cylinder is given below:

How to Find the Area of a Hollow Cylinder 

A hollow cylinder looks like a pipe or a tube. It has two circular ends – an outer circle and an inner circle – and a height. To find the total surface area (TSA) of a hollow cylinder, you need the external radius (R), internal radius (r), and height (h).

Let’s understand it better with an example.

Example:

Find the total surface area of a hollow cylinder with:

• External radius (R) = 10 cm
   
• Internal radius (r) = 6 cm
   
• Height (h) = 15 cm

Step 1: Use the formula

Total Surface Area (TSA) of a hollow cylinder is:
TSA = 2πh(R + r) + 2π(R² - r²)

Step 2:

 TSA = 2 × 3.1416 × 15 × (10 + 6) + 2 × 3.1416 × (10² - 6²)
TSA = 2 × 3.1416 × 15 × 16 + 2 × 3.1416 × (100 - 36)
TSA = 1507.968 + 402.123
TSA ≈ 1910.09 cm²

Important Points on Area of Hollow Cylinder

The important points on the area of a hollow cylinder are given below:

• There are four types of areas possible for a hollow cylinder.
• There are two radius for a hollow cylinder and thus two surfaces.
• Curved surface area for a hollow cylinder is the sum of the inner and outer surface area excluding the base area.
• The cross-section of a hollow cylinder is a ring.
• Total surface area is the sum of all different parts of the hollow cylinder.

Solved Examples of Area of Hollow Cylinder

Example 1: The lateral surface area of a hollow cylinder is 4224 \(cm^2\). If we cut this hollow cylinder along its height and form a rectangle of breadth 20 cm. Find the perimeter of the rectangular sheet.

Solution: Area of Rectangular Sheet = Surface Area of the Cylinder

Area of Rectangle = l x b
⇒ l x b = 4224
⇒ l x 20 = 4224
⇒ l = 211.2 cm

Perimeter of the rectangle = 2( l + b)
= 2 (211.2 + 20)
= 2 x 231.2 = 462.4 cm

Hence, the perimeter of the rectangle will be 462.4 cm.

Example 2: If the inner and outer radius of a water pipe is 2 cm and 7cm respectively, then find the total surface area of the pipe of length 30 cm.

Solution: According to the question we have, inner radius r = 2 cm, outer radius R = 7 cm and height h = 30 cm. The length of the pipe is the height of the pipe if we imagine the pipe vertically.

\( \begin{array}{l}TSA=2\pi\left(R+r\right)h+2\pi\left(R^2-r^2\right)\\\ \ \ \ \ \ \ =2\pi\left(7+2\right)30+2\pi\left(\left(7\right)^2-\left(2\right)^2\right)\\\ \ \ \ \ \ \ =2\pi\left(9\right)30+2\pi\left(45\right)\\\ \ \ \ \ \ \ =\left(2\times3.14\times9\times30\right)+\left(2\times3.14\times45\right)\\\ \ \ \ \ \ \ =1695.6+282.6\\\ \ \ \ \ \ \ =1978.2\end{array}\)

Thus, the total surface area for the given water pipe is \( 1978.2\ cm^2 \)

Example 3: Calculate the curved surface area for a hollow cylinder whose outer radius is 3.6 mm and inner radius is 0.9 mm. Also, the cylinder has a height of 100 mm.

Solution: So we have r = 0.9 mm and R = 3.6 mm and h = 100 mm

\( \begin{array}{l}CSA=2\pi\left(r+R\right)h\\\ \ \ \ \ \ \ =2\pi\left(0.9+3.6\right)100\\\ \ \ \ \ \ \ =200\pi\left(4.5\right)\\\ \ \ \ \ \ \ =200\times3.14\times4.5\\\ \ \ \ \ \ \ =2826\end{array}\)

Thus we the required curved surface area of the given hollow cylinder as \( 2826\ cm^2 \)

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