To find the surface area of the triangular prism, we need to calculate the area of the triangular base and the areas of the rectangular faces.
Step 1: Find the area of the triangular base
The triangular base is a right triangle with the following dimensions:
- Height (left side) = 8 cm
- Base = 6 cm
The area of a triangle is given by the formula: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] Substituting the values: \[ \text{Area} = \frac{1}{2} \times 6 \times 8 = \frac{1}{2} \times 48 = 24 \text{ cm}^2 \]
Step 2: Find the surface area of the three rectangular faces
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Front Rectangular Face: This face has dimensions of 6 cm (base of the triangle) * 9 cm (width of the prism). \[ \text{Area} = 6 \times 9 = 54 \text{ cm}^2 \]
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Left Rectangular Face: This face has dimensions of 8 cm (height of the triangle) * 9 cm (width of the prism). \[ \text{Area} = 8 \times 9 = 72 \text{ cm}^2 \]
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Right Rectangular Face: The length of this face corresponds to the hypotenuse of the triangle. The hypotenuse measures 10 cm. \[ \text{Area} = 10 \times 9 = 90 \text{ cm}^2 \]
Step 3: Calculate the total surface area
The total surface area of the prism is the sum of the areas of the two triangular bases and the three rectangular faces. Since the prism has two triangular bases, we multiply the area of the triangular base by 2.
So the total surface area is: \[ \text{Total Surface Area} = 2 \times \text{Area of Triangle} + \text{Area of Front Rectangular Face} + \text{Area of Left Rectangular Face} + \text{Area of Right Rectangular Face} \]
Calculating: \[ \text{Total Surface Area} = 2 \times 24 + 54 + 72 + 90 \] \[ \text{Total Surface Area} = 48 + 54 + 72 + 90 \] \[ \text{Total Surface Area} = 264 \text{ cm}^2 \]
Final answer:
The surface area of the triangular prism is 264 square centimeters.