Surface Area of Composite 3D Figures Practice

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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)
m2

Answer 1

To find the total surface area of the composite 3D figure, we need to calculate the surface area of both the rectangular prism and the triangular prism, and then combine them while accounting for the overlapping areas.

Step 1: Surface Area of the Rectangular Prism

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

• \(l = 32 , \text{m}\) (length)
• \(w = 10 , \text{m}\) (width)
• \(h = 8 , \text{m}\) (height)

Calculating each area:

• \(lw = 32 \times 10 = 320 , \text{m}^2\)
• \(lh = 32 \times 8 = 256 , \text{m}^2\)
• \(wh = 10 \times 8 = 80 , \text{m}^2\)

Now substitute these values into the surface area formula: \[ SA_{\text{rectangular}} = 2(320 + 256 + 80) = 2(656) = 1312 , \text{m}^2 \]

Step 2: Surface Area of the Triangular Prism

The formula for the surface area of a triangular prism is: \[ SA = bh + \text{Perimeter of the base} \times h \] Where:

• \(b\) is the area of the triangular base
• \(h\) is the height of the prism (8 m for the rectangular prism above)

Area of the Triangular Base:

The triangular base is a right triangle with:

• Base (\(b = 10\) m, width of the rectangular prism)
• Height (\(h = 12\) m)

Calculating the area: \[ \text{Area}_{\text{triangle}} = \frac{1}{2} \times b \times h = \frac{1}{2} \times 10 \times 12 = 60 , \text{m}^2 \]

Perimeter of the Triangular Base:

The perimeter includes the two sides (20 m each) and the base (10 m): \[ \text{Perimeter} = 20 + 20 + 10 = 50 , \text{m} \]

Total Surface Area of the Triangular Prism:

\[ SA_{\text{triangular}} = \text{Area}{\text{triangle}} + \text{Perimeter} \times h \] \[ SA{\text{triangular}} = 60 + 50 \times 8 = 60 + 400 = 460 , \text{m}^2 \]

Step 3: Combine the Surface Areas

Now, combine the two surface areas but subtract the area where the triangular prism sits on the rectangular prism because those areas are not exposed:

• Area of the base of the triangular prism that is on the rectangular prism (the triangular base area): \[ \text{Area}_{\text{triangle base}} = 60 , \text{m}^2 \]

Total surface area of the composite figure: \[ SA_{\text{total}} = SA_{\text{rectangular}} + SA_{\text{triangular}} - \text{Area}{\text{triangle base}} \] \[ SA{\text{total}} = 1312 + 460 - 60 = 1712 , \text{m}^2 \]

Final Answer

The total surface area of the composite figure is: \[ \boxed{1712} , \text{m}^2 \]