To find the surface area of a triangular prism, we need to determine the areas of the two triangular bases and the three rectangular faces that connect those bases.
The dimensions you provided are:
- Two sides of the triangular base are 6 cm and 8 cm.
- The height of the triangular base is 9 cm.
- The length of the prism (the distance between the two triangular bases) is 10 cm.
First, we find the area of the triangular base:
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Area of triangle: The area \(A\) of a triangle can be calculated using the formula: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \] We can take 6 cm as the base and 9 cm as the height. \[ A = \frac{1}{2} \times 6 \times 9 = 27 \text{ cm}^2 \]
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Area of triangular bases: Since there are two triangular bases, \[ \text{Area of two triangles} = 2 \times 27 = 54 \text{ cm}^2 \]
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Area of rectangular faces: The prism has three rectangular faces, with dimensions:
- One rectangle will have a length of 10 cm (the height of the prism) and a width of 6 cm. Its area is: \[ A_1 = 10 \times 6 = 60 \text{ cm}^2 \]
- Another rectangle will have a length of 10 cm and a width of 8 cm. Its area is: \[ A_2 = 10 \times 8 = 80 \text{ cm}^2 \]
- The third rectangle will have a length of 10 cm and can be found using the Pythagorean theorem to determine the third side (let's assume the triangle is not a right triangle). If we denote the third side as \(a\), we can find the height first.
However, if we only use the two given sides for the prism, we may lack this information.
For completeness, let’s find the area assuming the triangles are right-angled. Calculating without needing to calculate the exact side directly, we can assume if we did find the third side, we could express everything related to it. Otherwise, you might need to clarify or provide this to get an accurate calculation.
Adding it all together: \[ \text{Total area} = \text{Area of two triangular bases} + \text{Areas of rectangular faces} \]
Calculating specifically: If we simply assume the third rectangle uses either of the triangle sides in the same logic, the overall surface area will fully leverage \(A_3\) based on it.
Assuming the values given summarize as: \[ 3D = 54 + 60 + 80 : + A_3 \]
Thus, to finalize the surface area from these calculations, we arrive at a specific value once all sides' contributions are adequately accounted.
In summary, kindly clarify how edge lanes among 6, 8, 10, and needed height apply overall. Post this: \[ \text{Surface Area} = 54 + 60 + 80 + a \text{(sum total rimmed)} \] Final values once exact sides are identified for full resolution.