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Find the surface area of this pyramid.

To find the surface area of a pyramid, we need to calculate both the lateral surface area and the base area.

Given:
- Height of the pyramid from the base to the apex: 12 meters
- Apothem of the base (distance from the center to the midpoint of a side): 6 meters
- Side length of the base: 8 meters

The base of the pyramid is a regular polygon (in this case, it seems to be a square since the side length is provided).

### Step 1: Calculate the Base Area
The base is a square with side length \( s = 8 \, \text{meters} \).

\[ \text{Base Area} = s^2 = 8^2 = 64 \, \text{square meters} \]

### Step 2: Calculate the Slant Height
The slant height can be found using the Pythagorean theorem in the triangle formed by the height of the pyramid, the apothem of the base, and the slant height.

The apothem is half of the base side length for a square, which we are given directly as 6 meters.

\[ l = \sqrt{(12^2 + 6^2)} = \sqrt{(144 + 36)} = \sqrt{180} = 6\sqrt{5} \, \text{meters} \]

### Step 3: Calculate the Lateral Surface Area
The lateral surface area for a pyramid is calculated as:

\[ \text{Lateral Surface Area} = \frac{1}{2} \times \text{Perimeter of the base} \times \text{Slant height} \]

The perimeter of the base \( P \) for a square is:

\[ P = 4 \times s = 4 \times 8 = 32 \, \text{meters} \]

So the lateral surface area \( A_{\text{lateral}} \) is:

\[ A_{\text{lateral}} = \frac{1}{2} \times 32 \times 6\sqrt{5} = 16 \times 6\sqrt{5} = 96\sqrt{5} \, \text{square meters} \]

### Step 4: Total Surface Area
Adding the base area and the lateral surface area:

\[ \text{Total Surface Area} = \text{Base Area} + \text{Lateral Surface Area} \]
\[ \text{Total Surface Area} = 64 + 96\sqrt{5} \, \text{square meters} \]

If you prefer the numerical value, you can approximate \(\sqrt{5} \approx 2.236\):

\[ \text{Lateral Surface Area} \approx 96 \times 2.236 = 214.656 \, \text{square meters} \]

\[ \text{Total Surface Area} \approx 64 + 214.656 = 278.656 \, \text{square meters} \]

Thus, the surface area of the pyramid is approximately 278.656 square meters.

Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. How do you calculate the volume of this pyramid?
2. What is the formula for the height of a pyramid if only the slant height and base side length are known?
3. How would the surface area change if the height of the pyramid were doubled?
4. Can you explain how the lateral surface area of a pyramid is derived?
5. How do you convert the surface area of the pyramid into square feet?

Tip: Always double-check the measurements and ensure the correct formulas are applied, especially distinguishing between height, slant height, and apothem.

Learn how to find the surface area of a pyramid with a height of 12 meters and a base side length of 8 meters. Includes step-by-step calculations of the base area, slant height, lateral surface area, and total surface area. Suitable for students in Grades 9-12 studying geometry and pyramids.

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