To calculate the surface area of a 3D triangular prism, you'll need to consider both the triangular bases and the three rectangular sides. Here's a step-by-step guide:
Step 1: Identify the dimensions
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Base Triangle: Determine the dimensions of the triangular base. Let:
- \( a \), \( b \), and \( c \) be the lengths of the sides of the triangle.
- \( h \) be the height of the triangle.
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Height of the Prism: Let \( H \) be the height (length) of the prism.
Step 2: Calculate the area of the triangular base
The area \( A_\triangle \) of the triangular base can be calculated using the formula: \[ A_\triangle = \frac{1}{2} \times \text{base} \times \text{height} \] You can also use Heron’s formula if you have all three sides: \[ s = \frac{a + b + c}{2} \quad (\text{semi-perimeter}) \] \[ A_\triangle = \sqrt{s(s - a)(s - b)(s - c)} \]
Step 3: Calculate the areas of the rectangular sides
The prism has three rectangular sides. The areas of these rectangles are calculated as follows:
- For the side corresponding to side \( a \): \[ A_a = a \times H \]
- For the side corresponding to side \( b \): \[ A_b = b \times H \]
- For the side corresponding to side \( c \): \[ A_c = c \times H \]
Step 4: Calculate the total surface area
The total surface area \( SA \) of the triangular prism comprises:
- The areas of the two triangular bases (there are two bases): \[ 2 \times A_\triangle \]
- The areas of the three rectangular sides: \[ A_a + A_b + A_c \]
Thus, the total surface area can be expressed as: \[ SA = 2 \times A_\triangle + A_a + A_b + A_c \]
Putting it all together
The complete formula for the surface area of a triangular prism becomes: \[ SA = 2 \times A_\triangle + (a + b + c) \times H \]
Make sure to substitute the appropriate values to get your final surface area!
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