Jennifer got a box of chocolates. The box is a right triangular prism shaped box. It is 7 inches long, and the triangular base measures 2 in x 3 in x 4 in. What is the surface area of the box of chocolates?

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  1. The easy part is to find the areas of the 3 rectangles, they are:
    2x7, 3x7, and 4x7.

    The hard part is to find the area of the triangular base.
    Since it is not right-angled, 2^2 + 3^2 ≠ 4^2, we need more complex methods.

    method one: Heron's formula
    area = √(s(s-a)(s-b)(s-c)), where s is 1/2 the perimeter, and a, b, and c are the three sides.
    area = √(4.5 * 2.5 * 1.5 * .5) = appr 2.905

    method two: Use the cosine law to find one of the angles Ø, then area = (1/2)(a)(b)sinØ, where Ø is the contained angle between sides a and b.
    I will pick the angle opposite side 2
    2^2 = 3^2 + 4^2 - 2(3)(4)cosØ
    cosØ = 21/24 = ....
    Ø = 28.955°
    area = (1/2)(4)(3)sin28.955 = appr 2.905 , same as before

    method 3
    make a sketch with 4 as the base, draw an altitude to meet the base, call it h. Let the base be cut into parts x, and 4-x
    You now have 2 right-angled triangles:
    x^2 + h^2 = 3^2 and (4-x)^2 + h^2 = 2^2

    solve for h and use (1/2)(4)(h), you must get 2.905

    Add up your three rectangles plus your two triangles areas.

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  2. what is the answer

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