https://file.aiquickdraw.com/mathbot/file/1724432067189aw81lyz0.jpg

To calculate the total surface area of the cylinder shown in the image, we'll use the formula for the surface area of a cylinder:

\[
A_{\text{total}} = 2\pi r h + 2\pi r^2
\]

Where:
- \( r \) is the radius of the base,
- \( h \) is the height of the cylinder,
- \( \pi \) is a mathematical constant approximately equal to 3.14159.

From the image:
- The radius \( r \) is given as 8 cm.
- The height \( h \) is given as 15 cm.

Let's calculate:

1. **Lateral Surface Area** \( A_{\text{lateral}} \):
\[
A_{\text{lateral}} = 2\pi rh = 2\pi (8)(15) = 240\pi \, \text{cm}^2
\]

2. **Area of the Two Bases** \( A_{\text{bases}} \):
\[
A_{\text{bases}} = 2\pi r^2 = 2\pi (8^2) = 128\pi \, \text{cm}^2
\]

3. **Total Surface Area** \( A_{\text{total}} \):
\[
A_{\text{total}} = 240\pi + 128\pi = 368\pi \, \text{cm}^2
\]

Finally, to get the numerical value:
\[
A_{\text{total}} \approx 368 \times 3.14159 = 1156.64 \, \text{cm}^2
\]

So, the total surface area of the cylinder is approximately **1156.64 cm²**.

Would you like more details or have any questions? 

Here are some related questions to consider:
1. What is the formula for the volume of a cylinder, and how would you calculate it for this cylinder?
2. How does changing the radius while keeping the height constant affect the total surface area?
3. If the cylinder's height were doubled, what would be the new total surface area?
4. How would the surface area change if the radius were increased by 50%?
5. What is the importance of understanding surface area in real-world applications?

**Tip:** When dealing with problems involving cylinders, always remember to calculate both the lateral surface area and the area of the bases separately before summing them for the total surface area.

Learn how to calculate the total surface area of a cylinder using the formula A_{\text{total}} = 2\pi r h + 2\pi r^2. In this example, the radius is 8 cm and the height is 15 cm, resulting in a surface area of approximately 1156.64 cm². Explore related questions on cylinder volume, changing dimensions, and real-world applications.

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