Calculate the surface area of a triangular prism.
Understand the Problem
The question is asking us to calculate the surface area of a triangular prism, which typically involves finding the areas of the triangular bases and the three rectangular sides, and summing them up.
Answer
$$ SA = 2 \times \left(\frac{1}{2} \times b \times h\right) + b \times L + s_2 \times L + s_3 \times L $$
Answer for screen readers
The total surface area of the triangular prism is given by the formula: $$ SA = 2 \times \left(\frac{1}{2} \times b \times h\right) + b \times L + s_2 \times L + s_3 \times L $$
Steps to Solve
- Identify the dimensions of the prism
First, gather the dimensions of the triangular prism: the base length ($b$), the height of the triangle ($h$), and the length or height of the prism ($L$).
- Calculate the area of the triangular base
The area of a triangle is calculated using the formula: $$ A_{triangle} = \frac{1}{2} \times b \times h $$
- Calculate the area of the rectangular sides
Next, calculate the areas of the three rectangular sides. The areas are calculated as follows:
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The first rectangle (opposite the base of the triangle) has an area of: $$ A_1 = b \times L $$
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The second rectangle (one of the other sides) is calculated using the triangle's second side length ($s_2$) as: $$ A_2 = s_2 \times L $$
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The third rectangle (the last side) is calculated using the triangle's third side length ($s_3$) as: $$ A_3 = s_3 \times L $$
- Sum the areas to find the total surface area
Finally, sum the areas of the triangular bases and the three rectangles to find the total surface area: $$ SA = 2 \times A_{triangle} + A_1 + A_2 + A_3 $$
- Plug in the dimensions and calculate
Replace $b$, $h$, $L$, $s_2$, and $s_3$ with their actual values to compute the total surface area $SA$.
The total surface area of the triangular prism is given by the formula: $$ SA = 2 \times \left(\frac{1}{2} \times b \times h\right) + b \times L + s_2 \times L + s_3 \times L $$
More Information
The surface area of a triangular prism takes into account both the triangular bases and the three rectangular sides. It's important in applications like packaging design and construction where knowing the surface area informs material requirements.
Tips
- Forgetting to multiply by 2 for the triangular bases.
- Mixing up or miscalculating the side lengths of the triangle when determining the areas of the rectangular sides.
- Omitting any one of the rectangular sides in the final sum.
