Construct nets of prisms and pyramids

A net represents a prism or pyramid as a connected arrangement of its two-dimensional faces that can be folded to form the three-dimensional solid, with matching edge lengths joined and no faces overlapping. Understanding includes identifying the congruent bases and lateral faces, reasoning that multiple valid nets may represent the same solid, and using the net to interpret or determine surface area; more complex polyhedra and generalized net classifications are not included.

Detailed Explanation: Construct nets of prisms and pyramids

A net is a flat arrangement of all the faces of a solid. When the faces are folded along their joined edges, they form the 3-D shape.

For a prism:

There are two congruent bases.
The side faces are rectangles.
Each rectangle’s width matches one side of the base.
The rectangles must be connected so they can wrap around the bases.

Example: Make a net of a triangular prism

The triangular base has side lengths $3$ , $4$ , and $5$ units. The prism is $8$ units long.

Step 1: Identify the faces

The prism has:

Two congruent triangles, each with side lengths $3$ , $4$ , and $5$ .
Three rectangles, one for each side of the triangle.

Since the prism is $8$ units long, the rectangles are:

8\times3,\qquad 8\times4,\qquad 8\times5

Step 2: Draw the rectangles

Join the three rectangles along their $8$ -unit edges to make a strip:

8\times3 \quad\text{next to}\quad 8\times4 \quad\text{next to}\quad 8\times5

The rectangles can be arranged in a different order, as long as the side lengths match correctly.

Step 3: Add the triangular bases

Draw two congruent triangles with side lengths $3$ , $4$ , and $5$ .

Attach each triangle to a $3$ -unit edge of the $8\times3$ rectangle, one triangle at each end of that rectangle. The triangles should face outward so that they will become the two ends of the prism when folded.

A description of the net is:

        triangle
            |
[triangle] [8 × 3]—[8 × 4]—[8 × 5]
            |
        triangle

The exact placement can look different, but every rectangle must connect to the rectangles beside it, and the triangle edges must match the rectangle widths.

Step 4: Check the net

Before folding, check that:

The two triangles are the same size and shape.
The $3$ -unit triangle side matches the width of the $8\times3$ rectangle.
The $4$ -unit triangle side matches the width of the $8\times4$ rectangle.
The $5$ -unit triangle side matches the width of the $8\times5$ rectangle.
No faces overlap in the flat net.

When folded, the three rectangles wrap around and the two triangles become the two bases.

The net can also help find surface area:

\text{Surface area} =2\left(\frac12\cdot3\cdot4\right) +(8\cdot3+8\cdot4+8\cdot5)

=12+24+32+40=108\text{ square units}

Learn by doing: Construct nets of prisms and pyramids

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Nets of 3D Shapes - Shape Image to Net

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