To find the lateral and surface area of a triangular prism, we need to know the dimensions of the prism and the area of its base.
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Base Area of the Triangular Prism: The triangular base has a base \( b = 6 \) ft and a height \( h = 4 \) ft. The area \( A \) of the triangle can be calculated using the formula: \[ A = \frac{1}{2} \times b \times h = \frac{1}{2} \times 6 \times 4 = 12 \text{ ft}^2 \]
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Lateral Area: The lateral area of a prism is the sum of the areas of the three rectangular faces connecting the triangular bases. The lateral area \( LA \) can be calculated using the formula: \[ LA = \text{Perimeter of base} \times \text{Height of prism} \] To find the perimeter of the base triangle, we need the lengths of all three sides. The given right length (hypotenuse) is 5 ft, and one leg is 4 ft (which corresponds to the height). The other leg, which can be calculated using the Pythagorean theorem, is as follows:
Let \( a \) be the leg opposite the 4 ft side: \[ a^2 + 4^2 = 5^2 \ a^2 + 16 = 25 \ a^2 = 9 \ a = 3 \text{ ft} \]
Now the lengths of the sides of the triangle are 6 ft, 4 ft, and 5 ft. The perimeter \( P \) of the triangular base is: \[ P = 6 + 4 + 5 = 15 \text{ ft} \]
Now we can calculate the lateral area: \[ LA = P \times \text{Height} = 15 \text{ ft} \times 8 \text{ ft} = 120 \text{ ft}^2 \]
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Surface Area: The surface area \( SA \) of the prism is the sum of the lateral area and the area of the two triangular bases: \[ SA = LA + 2 \times \text{Area of triangular base} \ SA = 120 \text{ ft}^2 + 2 \times 12 \text{ ft}^2 = 120 \text{ ft}^2 + 24 \text{ ft}^2 = 144 \text{ ft}^2 \]
Summary of Results:
- Lateral Area: \( 120 \text{ ft}^2 \)
- Surface Area: \( 144 \text{ ft}^2 \)