To find the surface area of a triangular prism, we need to calculate the area of the triangular base and the lateral surface area.
Given the dimensions:
- Sides of the triangular base: \( a = 8 \) cm, \( b = 8 \) cm, \( c = 6.9 \) cm
- Height of the prism (length): \( h = 6 \) cm (assuming this is the height of the prism, as the problem does not specify)
Step 1: Calculate the area of the triangular base using Heron's formula.
First, we find the semi-perimeter \( s \) of the triangle:
\[ s = \frac{a + b + c}{2} = \frac{8 + 8 + 6.9}{2} = \frac{22.9}{2} = 11.45 , \text{cm} \]
Now, we can apply Heron's formula, which is:
\[ \text{Area} = \sqrt{s(s - a)(s - b)(s - c)} \]
Calculating each term:
\[ s - a = 11.45 - 8 = 3.45 , \text{cm} \] \[ s - b = 11.45 - 8 = 3.45 , \text{cm} \] \[ s - c = 11.45 - 6.9 = 4.55 , \text{cm} \]
Now plug these values into Heron's formula:
\[ \text{Area} = \sqrt{11.45 \times 3.45 \times 3.45 \times 4.55} \]
Now performing the multiplications:
\[ 11.45 \times 3.45 \approx 39.4525 \] \[ 39.4525 \times 3.45 \approx 136.304125 \] \[ 136.304125 \times 4.55 \approx 620.3801875 \]
Taking the square root:
\[ \text{Area} \approx \sqrt{620.3801875} \approx 24.9 , \text{cm}^2 \]
Step 2: Calculate the lateral surface area of the prism.
The lateral surface area is given by the perimeter of the triangular base multiplied by the height of the prism.
First, calculate the perimeter \( P \) of the triangular base:
\[ P = a + b + c = 8 + 8 + 6.9 = 22.9 , \text{cm} \]
Now, find the lateral surface area:
\[ \text{Lateral Surface Area} = P \times h = 22.9 \times 6 = 137.4 , \text{cm}^2 \]
Step 3: Calculate the total surface area.
The total surface area \( SA \) of the prism is the area of the two triangular bases plus the lateral surface area:
\[ SA = 2 \times \text{Area of one triangular base} + \text{Lateral Surface Area} \] \[ = 2 \times 24.9 + 137.4 \] \[ = 49.8 + 137.4 = 187.2 , \text{cm}^2 \]
Final Answer
The surface area of the triangular prism is approximately \( \boxed{187.2} , \text{cm}^2 \).