A package is in the shape of a triangular prism. The bases are right triangles with perpendicular legs measuring 9 centimeters and 12 centimeters. The distance between the bases is 10 centimeters. What is the surface area of the triangular prism? square centimeters

The surface area of the triangular prism is 468 (square cm).

The surface area A of a triangular prism is given by the formula:

[tex]\[ A = 2A_{\text{base}} + P_{\text{base}} \times h \][/tex]

where:

- [tex]\( A_{\text{base}} \)[/tex] is the area of one of the triangular bases,

- [tex]\( P_{\text{base}} \)[/tex] is the perimeter of one of the triangular bases,

- h is the distance between the bases.

The area of a right triangle is given by:

[tex]\[ A_{\text{base}} = \frac{1}{2} \times \text{base} \times \text{height} \][/tex]

And the perimeter of a right triangle is the sum of the lengths of its three sides.

Given that the legs of the right triangles forming the bases are 9 cm and 12 cm, and the distance between the bases is 10 cm, we can calculate the surface area.

1. Area of the triangular base [tex](\(A_{\text{base}})\)[/tex]:

[tex]\[ A_{\text{base}} = \frac{1}{2} \times 9 \times 12 \][/tex]

2. Perimeter of the triangular base [tex](\(P_{\text{base}})\)[/tex]:

[tex]\[ P_{\text{base}} = 9 + 12 + \sqrt{9^2 + 12^2} \][/tex]

3. Surface area of the triangular prism A:

[tex]\[ A = 2 \times A_{\text{base}} + P_{\text{base}} \times 10 \][/tex]

Calculate each part and find the total surface area.

[tex]\[ A_{\text{base}} = \frac{1}{2} \times 9 \times 12 = 54 \, \text{cm}^2 \][/tex]

[tex]\[ P_{\text{base}} = 9 + 12 + \sqrt{9^2 + 12^2} = 9 + 12 + 15 = 36 \, \text{cm} \][/tex]

[tex]\[ A = 2 \times 54 + 36 \times 10 = 108 + 360 = 468 \, \text{cm}^2 \][/tex]

So, the surface area of the triangular prism is [tex]\(468 \, \text{cm}^2\)[/tex].

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