To find the lateral surface area of an equilateral triangular prism, we need to understand the structure of the prism and how the lateral surfaces are formed.
An equilateral triangular prism consists of:
- Two triangular bases (top and bottom).
- Three rectangular lateral faces.
Given dimensions:
- The base length of the triangular base is \(6 , \text{cm}\) (the bottom edge of the triangle).
- The height of the triangular base is approximately \(4.5 , \text{cm}\).
- The height (length) of the prism, which connects the two triangular bases, is \(9 , \text{cm}\).
Step 1: Calculate the lateral surface area
The lateral surface area \(A_L\) of the prism can be computed as the sum of the areas of the three rectangular lateral faces.
Each lateral face is formed by extending the sides of the triangular base along the height of the prism. Here's how we can calculate the area of each lateral face:
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Identify the lengths of the sides of the triangular base:
- In an equilateral triangle, all sides are equal to \(6 , \text{cm}\). Hence, the three sides are: \(6 , \text{cm}\), \(6 , \text{cm}\), and \(6 , \text{cm}\).
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Calculate the area of the rectangular faces:
- Each rectangle's area can be calculated using the formula: \[ \text{Area of rectangle} = \text{side length} \times \text{height of prism} \]
For the three rectangles:
- First rectangle (side length \(6 , \text{cm}\)): \[ A_1 = 6 , \text{cm} \times 9 , \text{cm} = 54 , \text{cm}^2 \]
- Second rectangle (side length \(6 , \text{cm}\)): \[ A_2 = 6 , \text{cm} \times 9 , \text{cm} = 54 , \text{cm}^2 \]
- Third rectangle (side length \(6 , \text{cm}\)): \[ A_3 = 6 , \text{cm} \times 9 , \text{cm} = 54 , \text{cm}^2 \]
Step 2: Add all the rectangular areas
Now we can find the total lateral surface area \(A_L\):
\[ A_L = A_1 + A_2 + A_3 = 54 , \text{cm}^2 + 54 , \text{cm}^2 + 54 , \text{cm}^2 = 162 , \text{cm}^2 \]
Conclusion
The lateral surface area of the equilateral triangular prism is:
\[ \boxed{162 , \text{cm}^2} \]