Question

understanding nets of composite figures
the top two nets represent the triangular prisms that are attached to the bases of a rectangular prism. the composite figure is composed of all three prisms. the shaded areas are faces that are shared between the prisms. what is the total surface area of the composite figure?
78 square units
342 square units
282 square units
144 square units

Explanation:

Step1: Calculate the surface area of one triangular prism

• Area of the triangular base: \(A_{triangle}=\frac{1}{2}\times4\times3 = 6\)
• Perimeter of the triangular base: \(P = 3 + 4+5=12\)
• Lateral surface area of the triangular prism: \(A_{lateral}=12\times10 = 120\)
• Total surface area of one triangular prism: \(A_{tri - prism}=2\times6+120=132\)

Step2: Calculate the surface area of the rectangular prism

• The formula for the surface area of a rectangular prism \(A_{rect - prism}=2(lw+lh+wh)\)
• Here \(l = 10\), \(w = 3\), \(h = 3\)
• \(A_{rect - prism}=2(10\times3 + 10\times3+3\times3)=2(30 + 30+9)=2\times69 = 138\)

Step3: Calculate the total surface area of the composite figure

• Since the shaded areas (two rectangles of area \(3\times10\) each) are shared
• Total surface area \(A=2\times132+138-2\times(3\times10)\)
• \(A = 264+138 - 60\)
• \(A=342\)

Answer:

342 square units