Question

problem 2
a. what polyhedron can be assembled from this net? explain how you know.
b. find the surface area of this polyhedron. show your reasoning.

Explanation:

a.

A polyhedron is a 3 - D shape with flat faces. The given net has two congruent trapezoidal bases (the top and bottom - like parts of the net when folded) and four rectangular lateral faces. A trapezoidal prism has two trapezoidal bases and four rectangular lateral faces. When we fold the net, the trapezoidal parts will form the bases and the rectangular parts will form the lateral faces.

Step1: Calculate the area of the trapezoidal bases

The formula for the area of a trapezoid is \(A=\frac{(a + b)h}{2}\), where \(a\) and \(b\) are the lengths of the parallel sides and \(h\) is the height.
For the trapezoid: \(a = 3\), \(b=5\), \(h = 2\)
\(A_{trapezoid}=\frac{(3 + 5)\times2}{2}=8\)
Since there are \(2\) trapezoidal bases, \(A_{bases}=2\times8 = 16\)

Step2: Calculate the area of the rectangular faces

There are \(4\) rectangular faces.

• Two rectangles with dimensions \(2\times5\): \(A_{1}=2\times(2\times5)=20\)
• Two rectangles with dimensions \(2\times3\): \(A_{2}=2\times(2\times3)=12\)
• Two rectangles with dimensions \(2\times4\): \(A_{3}=2\times(2\times4)=16\)

The total area of the rectangular faces \(A_{rectangles}=20 + 12+16=48\)

Step3: Calculate the total surface area

The surface area \(S\) of the trapezoidal prism is the sum of the area of the bases and the area of the lateral faces.
\(S=A_{bases}+A_{rectangles}\)
\(S = 16+48\)

Answer:

A trapezoidal prism.

b.

Assume the side - length of each square in the grid is \(1\) unit.