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An illustration shows a 3 D triangular prism with the triangular face as a right triangle. The edges that are not visible are represented by dashed lines. The length and base of the triangular prism measures 30.5 centimeters and 27.9 centimeters. The perpendicular side of the triangular face measures 5.1 centimeter and the hypotenuse measures 28.4 centimeter.

A toy came in a container that is shaped like a triangular prism. How much wrapping paper is needed to wrap the toy? Round your answer to the nearest whole number.

(1 point)
S.A.=
cm2

1 answer

To find the surface area of a triangular prism, we need to calculate the areas of the two triangular faces and the three rectangular faces.

  1. Calculate the area of the triangular face:

    • The formula for the area of a right triangle is: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
    • Here, the base (b) is 27.9 cm and the height (h) is 5.1 cm.
    • Thus, the area of one triangular face is: \[ \text{Area} = \frac{1}{2} \times 27.9 \times 5.1 = \frac{1}{2} \times 142.29 = 71.145 \text{ cm}^2 \]

  2. Total area of the two triangular faces: \[ \text{Total area of triangular faces} = 2 \times 71.145 = 142.29 \text{ cm}^2 \]

  3. Calculate the areas of the rectangular faces:

    • The three rectangular faces are formed by:

      • The rectangular face with base as the length of the prism and height as the side of the triangle.
      • The other two faces will use the sides of the triangular face (the legs of the triangle).

    • Length of the prism: \( l = 30.5 \) cm.

    • The sides of the triangle are:

      • One leg = \( 5.1 \) cm (the height).
      • The other leg = \( 27.9 \) cm (the base).

    • Area of the first rectangular face (with base as the height of the triangle): \[ \text{Area}_1 = \text{height} \times \text{length} = 5.1 \times 30.5 = 155.55 \text{ cm}^2 \]

    • Area of the second rectangular face (with base as the base of the triangle): \[ \text{Area}_2 = 27.9 \times 30.5 = 851.45 \text{ cm}^2 \]

    • Area of the rectangular face (hypotenuse): \[ \text{Area}_3 = 28.4 \times 30.5 = 867.2 \text{ cm}^2 \]

  4. Total area of the rectangular faces: \[ \text{Total area of rectangular faces} = \text{Area}_1 + \text{Area}_2 + \text{Area}_3 = 155.55 + 851.45 + 867.2 = 1874.2 \text{ cm}^2 \]

  5. Total Surface Area: \[ \text{Total Surface Area} = \text{Total area of triangular faces} + \text{Total area of rectangular faces} \] \[ \text{Total Surface Area} = 142.29 + 1874.2 = 2016.49 \text{ cm}^2 \]

Rounding to the nearest whole number gives: \[ \text{S.A.} = 2016 \text{ cm}^2 \]

Therefore, the amount of wrapping paper needed to wrap the toy is \( \boxed{2016} \) cm².