Calculating the surface area of a hollow cylinder involves understanding both its inner and outer surfaces. This guide will show you step-by-step how to find the surface area of a hollow cylinder using its inner and outer radii and height.
Step 1: Show the Surface Area Formula
The formula for the surface area \(SA\) of a hollow cylinder is:
\[ SA = 2 \cdot \pi \cdot h \cdot (R + r) + 2 \cdot \pi \cdot (R^2 - r^2) \]
Where:
- \(R\) is the outer radius of the cylinder.
- \(r\) is the inner radius of the cylinder.
- \(h\) is the height of the cylinder.
Step 2: Explain the Formula
In this formula:
- \(2 \cdot \pi \cdot h \cdot (R + r)\) represents the lateral surface area of both the inner and outer cylindrical surfaces.
- \(2 \cdot \pi \cdot (R^2 - r^2)\) represents the surface area of the two annular (ring-shaped) bases of the hollow cylinder.
The total surface area is the sum of the lateral surface area and the area of the annular bases.
Step 3: Insert Numbers as an Example
Let's consider a hollow cylinder with:
- Outer radius \(R = 7\) units
- Inner radius \(r = 5\) units
- Height \(h = 10\) units
Step 4: Calculate the Final Value
First, we substitute the values into the formula:
\[ SA = 2 \cdot \pi \cdot 10 \cdot (7 + 5) + 2 \cdot \pi \cdot (7^2 - 5^2) \]
Next, we simplify inside the parentheses:
\[ SA = 2 \cdot \pi \cdot 10 \cdot 12 + 2 \cdot \pi \cdot (49 - 25) \]
\[ SA = 2 \cdot \pi \cdot 120 + 2 \cdot \pi \cdot 24 \]
Now, multiply the numbers:
\[ SA = 240 \cdot \pi + 48 \cdot \pi \]
\[ SA = 288 \cdot \pi \]
For \(\pi \approx 3.14\):
\[ SA \approx 288 \cdot 3.14 \]
\[ SA \approx 904.32 \, \text{square units} \]
Final Value
The surface area of a hollow cylinder with an outer radius of 7 units, an inner radius of 5 units, and a height of 10 units is approximately 904.32 square units.
