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 area of the triangular faces

The area of a right - triangle is \(A=\frac{1}{2}\times base\times height\). For the triangle with \(base = 4\) and \(height=3\), \(A=\frac{1}{2}\times4\times3 = 6\) square units. There are \(4\) such triangular faces (two prisms, two triangles per prism), so \(4\times6=24\) square units.

Step2: Calculate the area of the non - shared rectangular faces of the triangular prisms

For the triangular prisms, the non - shared rectangular faces:

• One set of rectangles has dimensions \(5\times10\) (there are \(2\) such rectangles), \(A = 2\times5\times10=100\) square units.
• Another set of rectangles has dimensions \(4\times10\) (there are \(2\) such rectangles), \(A=2\times4\times10 = 80\) square units.

Step3: Calculate the area of the rectangular prism's non - shared faces

The rectangular prism (excluding the shared faces). The formula for the surface area of a rectangular prism is \(S = 2(lw+lh+wh)\). But we consider non - shared parts. The non - shared faces:

• There are \(2\) faces with dimensions \(3\times3\), \(A = 2\times3\times3=18\) square units.
• There are \(2\) faces with dimensions \(10\times3\) (excluding shared ones), \(A=2\times10\times3 = 60\) square units.

Step4: Sum up all the areas

$$24 + 100+80 + 18+60=282$$

Answer:

282 square units