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An illustration shows a three dimensional composite figure formed by a triangular prism stacked on top of a rectangular prism. The top, right, and front faces are visible. The faces and edges that are not visible are indicated by dashed lines. The rectangular prism has a length of 32 meters, a width of 10 meters, and a height of 8 meters. The triangular prism has a rectangular base that is aligned on all edges with the rectangular prism below. The perpendicular height of the triangular prism is marked by a right angle symbol from the top vertex to the center of the triangular face. The height of the triangular face of the prism measures 12 meters. The sides of the triangular faces of the triangular prism measure 20 meters.

What is the total surface area of the figure?

(1 point)

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1 answer

To find the total surface area of the composite figure consisting of a rectangular prism and a triangular prism stacked on top, we need to calculate the surface areas of both prisms and then combine them, taking care to exclude the areas where they overlap.

Step 1: Calculate the surface area of the rectangular prism.

The formula for the surface area \( SA \) of a rectangular prism is given by: \[ SA = 2(lw + lh + wh) \] where:

  • \( l \) = length
  • \( w \) = width
  • \( h \) = height

Given:

  • \( l = 32 , m \)
  • \( w = 10 , m \)
  • \( h = 8 , m \)

Substituting those values in: \[ SA = 2(32 \times 10 + 32 \times 8 + 10 \times 8) \] \[ = 2(320 + 256 + 80) \] \[ = 2(656) \] \[ = 1312 , m^2 \]

Step 2: Calculate the surface area of the triangular prism.

The formula for the surface area of a triangular prism is given by: \[ SA = bh + sl + 2A \] where:

  • \( b \) = base length
  • \( h \) = height of the prism
  • \( s \) = slant height
  • \( A \) = area of the triangular base

Triangular base calculations:

  1. Find the area of the triangular base:

    • The base of the triangular face is equal to the width of the rectangular prism, which is 10 m.
    • The height of the triangle is given as 12 m. \[ A = \frac{1}{2} \times b \times h = \frac{1}{2} \times 10 \times 12 = 60 , m^2 \]

  2. Calculate the slant height:

    • The slant height is given as 20 m.

  3. Find the total surface area: The heights for the triangular prism side area will involve the slant height and the height of the prism:

    • The height of the triangular prism is equal to the height of the rectangular prism, which is 8 m. \[ SA = (10 \times 8) + (20 \times 8) + (2 \times 60) \] \[ = 80 + 160 + 120 = 360 , m^2 \]

Step 3: Combine the surface areas, subtracting the overlapping area.

Since the top face of the rectangular prism is covered by the triangular prism, we need to subtract that area as well. The area of the top face of the rectangular prism is: \[ 10 \times 32 = 320 , m^2 \]

Final Calculation:

Now summing everything up: \[ \text{Total Surface Area} = SA_{\text{rectangular prism}} + SA_{\text{triangular prism}} - \text{area of the top face of rectangular prism} \] \[ = 1312 , m^2 + 360 , m^2 - 320 , m^2 \] \[ = 1312 + 360 - 320 = 1352 , m^2 \]

Therefore, the total surface area of the figure is: \[ \boxed{1352 , m^2} \]