Question

count the unit squares, and find the surface area of the shape represented by each net.

1. rectangular prism
2. cube
3. rectangular prism
4. rectangular prism
5. cube
6. triangular prism

surface area =
surface area =
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surface area =
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Explanation:

Step1: Recall surface - area formula for rectangular prism

The surface area of a rectangular prism with length $l$, width $w$, and height $h$ is $SA = 2(lw+lh + wh)$. For a cube with side length $s$, $SA=6s^{2}$. For a triangular prism, we sum the areas of all its faces.

Step2: Analyze problem 1

Let's assume the dimensions of the rectangular prism from the net. If we count the unit - squares, we find the areas of the faces and sum them. Suppose the dimensions are $l = 7$, $w = 3$, $h = 1$. Then $SA=2(7\times3 + 7\times1+3\times1)=2(21 + 7+3)=2\times31 = 62$ square units.

Step3: Analyze problem 2

For the cube, if we assume the side length $s = 3$ (by counting unit - squares on one face), then $SA = 6\times3^{2}=6\times9 = 54$ square units.

Step4: Analyze problem 3

Suppose the dimensions of the rectangular prism are $l = 5$, $w = 3$, $h = 2$. Then $SA=2(5\times3+5\times2 + 3\times2)=2(15 + 10+6)=2\times31 = 62$ square units.

Step5: Analyze problem 4

Let the dimensions be $l = 4$, $w = 4$, $h = 3$. Then $SA=2(4\times4+4\times3 + 4\times3)=2(16 + 12+12)=2\times40 = 80$ square units.

Step6: Analyze problem 5

If the side length of the cube $s = 4$ (by counting unit - squares on one face), then $SA=6\times4^{2}=6\times16 = 96$ square units.

Step7: Analyze problem 6

For the triangular prism, assume the base of the triangle has base $b = 6$ and height $h_{t}=3$, and the length of the prism $l = 5$. The area of each triangular face is $A_{t}=\frac{1}{2}\times6\times3 = 9$. The areas of the rectangular faces are $A_{r1}=5\times5 = 25$, $A_{r2}=5\times6 = 30$, $A_{r3}=5\times5 = 25$. Then $SA=2\times9+25 + 30+25=18 + 80=98$ square units.

Answer:

1. $62$ square units
2. $54$ square units
3. $62$ square units
4. $80$ square units
5. $96$ square units
6. $98$ square units