https://file.aiquickdraw.com/mathbot/file/172671401727425tdhtoj.jpg

The figure you've uploaded is a triangular pyramid (tetrahedron) with an equilateral triangle base, where each side of the triangle measures 8 cm. The height from the apex to the base is also given.

To find the surface area of the tetrahedron, we need to compute the area of its four triangular faces.

### Step 1: Calculate the area of the base
The base is an equilateral triangle with side length \( s = 8 \, \text{cm} \).

The area \( A_{\text{base}} \) of an equilateral triangle can be calculated using the formula:
\[
A_{\text{base}} = \frac{\sqrt{3}}{4} \cdot s^2
\]
Substitute \( s = 8 \):
\[
A_{\text{base}} = \frac{\sqrt{3}}{4} \cdot 8^2 = \frac{\sqrt{3}}{4} \cdot 64
\]
\[
A_{\text{base}} = 16\sqrt{3} \, \text{cm}^2
\]

### Step 2: Calculate the area of each lateral triangular face
Each lateral face is an isosceles triangle with a base of 8 cm and height (slant height) of 7 cm. The area \( A_{\text{lateral}} \) of one isosceles triangle is given by:
\[
A_{\text{lateral}} = \frac{1}{2} \cdot \text{base} \cdot \text{height} = \frac{1}{2} \cdot 8 \cdot 7 = 28 \, \text{cm}^2
\]

Since there are three identical lateral faces, the total lateral area is:
\[
A_{\text{lateral total}} = 3 \times 28 = 84 \, \text{cm}^2
\]

### Step 3: Calculate the total surface area
The total surface area of the tetrahedron is the sum of the base area and the lateral area:
\[
A_{\text{total}} = A_{\text{base}} + A_{\text{lateral total}} = 16\sqrt{3} + 84
\]
We can approximate \( \sqrt{3} \approx 1.732 \):
\[
A_{\text{total}} \approx 16 \times 1.732 + 84 = 27.712 + 84 = 111.712 \, \text{cm}^2
\]

Thus, the approximate surface area of the tetrahedron is:
\[
\boxed{111.71 \, \text{cm}^2}
\]

Would you like more details on any specific part of this calculation? Here are some related questions:

1. How do you calculate the volume of this tetrahedron?
2. Can the formula for the area of an equilateral triangle be derived?
3. What is the relationship between the slant height and the perpendicular height in a pyramid?
4. How would the surface area change if the slant height were different?
5. How do you generalize this approach for non-equilateral pyramids?

**Tip:** When dealing with pyramids or tetrahedrons, always remember to distinguish between the slant height and the perpendicular height.

This problem explores the surface area of a tetrahedron with equilateral triangle base, using the formula for triangle area and summing the areas of the four faces. It covers key geometric concepts such as calculating the area of an equilateral triangle and lateral triangular faces. Suitable for high school students (Grades 9-12).

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