To find the area of a hollow cylinder, you typically calculate its surface area. According to one reference, the surface area of a hollow cylinder is found using a specific formula.
A hollow cylinder is essentially a tube. It has an inner cylindrical surface and an outer cylindrical surface, and potentially two annular (ring-shaped) ends if it is closed.
According to one reference, a hollow cylinder is described as being made up of two thin sheets of rectangle having a length and breadth.
Key dimensions needed to calculate the area include:
- Outer Radius (r₁): The distance from the center to the outer edge of the cylinder.
- Inner Radius (r₂): The distance from the center to the inner edge of the cylinder.
- Height (h): The vertical distance between the top and bottom ends of the cylinder.
It is assumed that r₁ > r₂ for a solid wall.
The Surface Area Formula
The reference states that the surface area of the cylinder is equal to the surface area derived from its components, providing the following formula:
Surface Area = 2π( r1 + r2)( r1 – r2 +h)
This formula calculates the total surface area of a hollow cylinder that is closed at both the top and bottom ends.
What the Formula Represents
The formula 2π( r1 + r2)( r1 – r2 +h) accounts for all surfaces of a closed hollow cylinder: the outer wall, the inner wall, and the two ring-shaped ends.
Components of the Total Surface Area
The total surface area calculated by the formula is the sum of:
- Outer Lateral Surface Area: The area of the outer curved wall. This is calculated as 2πr₁h.
- Inner Lateral Surface Area: The area of the inner curved wall. This is calculated as 2πr₂h.
- Area of the Two Annular Ends: The area of the top and bottom rings. The area of a single ring is the area of the large circle minus the area of the small circle: πr₁² - πr₂². For two ends, this is 2 * (πr₁² - πr₂²) = 2π(r₁² - r₂²). Using the difference of squares factorization (a² - b² = (a-b)(a+b)), this area is 2π(r₁ - r₂)(r₁ + r₂).
Adding these components gives the total surface area:
Total Area = Outer Lateral Area + Inner Lateral Area + Area of Two Ends
Total Area = 2πr₁h + 2πr₂h + 2π(r₁² - r₂²)
Total Area = 2πh(r₁ + r₂) + 2π(r₁ - r₂)(r₁ + r₂)
Factoring out the common term 2π(r₁ + r₂):
Total Area = 2π(r₁ + r₂)[h + (r₁ - r₂)]
Total Area = 2π(r₁ + r₂)(h + r₁ - r₂)
Rearranging the terms inside the second parenthesis gives:
Total Area = 2π( r1 + r2)( r1 – r2 +h)
This confirms that the formula provided in the reference calculates the total surface area of a closed hollow cylinder.
How to Use the Formula
To find the total surface area of a closed hollow cylinder using this formula:
- Measure the outer radius (r₁).
- Measure the inner radius (r₂).
- Measure the height (h).
- Substitute these values into the formula 2π( r1 + r2)( r1 – r2 +h) and calculate the result.
Example:
Let's say you have a hollow cylinder with:
- Outer Radius (r₁) = 5 cm
- Inner Radius (r₂) = 3 cm
- Height (h) = 10 cm
Using the formula:
Surface Area = 2π (5 cm + 3 cm) (5 cm – 3 cm + 10 cm)
Surface Area = 2π (8 cm) (2 cm + 10 cm)
Surface Area = 2π (8 cm) (12 cm)
Surface Area = 192π cm²
If you approximate π as 3.14159, the surface area is approximately 192 * 3.14159 ≈ 603.186 cm².
Other Areas of a Hollow Cylinder
While the reference formula provides the total surface area of a closed hollow cylinder, you might also be interested in:
- Lateral Surface Area: This is the area of just the inner and outer walls (useful for open-ended pipes). It is found by 2πh(r₁ + r₂).
- Cross-Sectional Area: This is the area of the ring shape if you slice through the cylinder. It is found by π(r₁² - r₂²).
The formula provided in the reference specifically calculates the total surface area, including the ends.
