To find the surface area of a triangular prism, we will use the formula:
\[ \text{Surface Area} = \text{Base Area} + \text{Lateral Area} \]
1. Calculate the Area of the Triangular Base
The base is an isosceles triangle. For an isosceles triangle, the area can be calculated with:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
In this case, it appears that:
- The sides of the isosceles triangle are 35 cm and 37 cm (assuming 37 cm is the length of the equal sides and 35 cm is the base).
- The height of the triangle is determined using the Pythagorean theorem.
First, we need to find the height of the triangle. To find the height \( h \) from the top vertex to the base (which splits the base into two equal halves), we do the following calculations:
Let half the base be \( \frac{35}{2} = 17.5 , \text{cm} \). Now, we use the Pythagorean theorem:
\[ h^2 + (17.5)^2 = (37)^2 \]
Calculating \( 37^2 \) and \( (17.5)^2 \):
\[ 37^2 = 1369 \] \[ (17.5)^2 = 306.25 \]
Now substituting back into the equation:
\[ h^2 + 306.25 = 1369 \] \[ h^2 = 1369 - 306.25 = 1062.75 \] \[ h = \sqrt{1062.75} \approx 32.6 , \text{cm} \]
Now substitute back to find the area of the triangle:
\[ \text{Area} = \frac{1}{2} \times 35 , \text{cm} \times 32.6 , \text{cm} \approx 570.5 , \text{cm}^2 \]
2. Calculate the Lateral Area
The lateral area is calculated as the perimeter of the base triangle times the height of the prism:
- The perimeter \( P \) of the triangular base is:
\[ P = 35 + 37 + 37 = 109 , \text{cm} \]
- If the height (length) of the prism is 24 cm, then:
\[ \text{Lateral Area} = P \times \text{height} = 109 , \text{cm} \times 24 , \text{cm} = 2616 , \text{cm}^2 \]
3. Calculate the Total Surface Area
Now sum the areas:
\[ \text{Surface Area} = \text{Base Area} + \text{Lateral Area} = 570.5 , \text{cm}^2 + 2616 , \text{cm}^2 \approx 3186.5 , \text{cm}^2 \]
Conclusion
The total surface area of the triangular prism is approximately:
\[ \text{Surface Area} \approx 3186.5 , \text{cm}^2 \]
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