A cylindrical letterbox is 40cm high with a diameter of 20cm. Find the total surface area to estimate the amount of sheet metal it would take to construct the letterbox

To calculate the total surface area of the cylindrical letterbox, we need to consider both the lateral surface area (the side of the cylinder) and the surface area of the two circular ends (the top and bottom).

### 1. Formula for the total surface area of a cylinder:
\[
\text{Total Surface Area} = 2\pi r h + 2\pi r^2
\]
Where:
- \( r \) is the radius of the cylinder (half of the diameter),
- \( h \) is the height of the cylinder,
- \( \pi \) is approximately 3.1416.

### 2. Given dimensions:
- Height \( h = 40 \, \text{cm} \),
- Diameter \( d = 20 \, \text{cm} \), so radius \( r = \frac{d}{2} = 10 \, \text{cm} \).

### 3. Breakdown of the surface area:
#### a) Lateral surface area:
\[
2\pi r h = 2 \times 3.1416 \times 10 \times 40 = 2513.28 \, \text{cm}^2
\]
#### b) Area of the two circular ends:
\[
2\pi r^2 = 2 \times 3.1416 \times 10^2 = 628.32 \, \text{cm}^2
\]

### 4. Total surface area:
\[
\text{Total Surface Area} = 2513.28 + 628.32 = 3141.6 \, \text{cm}^2
\]

Thus, the total surface area of the letterbox is approximately **3141.6 cm²**.

Would you like further details on this process or have any questions?

### Follow-up questions:
1. How does the total surface area change if the height of the cylinder increases?
2. How would the surface area change if the diameter of the cylinder is doubled?
3. Can you calculate the surface area of a cylinder without using \(\pi\)?
4. How does the surface area of a cylinder compare to its volume?
5. How would you calculate the surface area of a hollow cylindrical structure?

### Tip:
Always remember that the total surface area of a cylinder includes both the side and the two circular bases unless otherwise stated.

A cylindrical letterbox is 40cm high with a diameter of 20cm. Find the total surface area to estimate the amount of sheet metal it would take to construct the letterbox.

Learn how to calculate the total surface area of a cylindrical letterbox with a height of 40cm and a diameter of 20cm. This problem involves geometry concepts like surface area of cylinders, using formulas for lateral surface area and the area of circular ends. Suitable for Grades 6-8.

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