Fold a rectangular-prism net and account for every outside face

Learning goal: Represent a rectangular prism with a net, identify matching edges and find its surface area by summing the areas of all six faces. Distinguish an unfolded outside surface from the volume enclosed by the folded solid. Nets that include triangular faces follow in the companion lesson.

A closed rectangular box has length four centimeters, width three centimeters and height two centimeters. Its six faces come in three pairs of congruent faces, faces with the same size and shape: two four-by-three rectangles, two four-by-two rectangles and two three-by-two rectangles. Opposite faces have equal dimensions even if only one of each pair is visible in a perspective drawing.

A net is an arrangement of connected flat faces that can fold into the solid without overlapping interiors or leaving a required face missing. The diagram places a four-by-three base in the center, attaches four side faces along its four edges and attaches a second four-by-three face to the outer edge of one four-by-two side. Folding the side faces upward and the final face over closes the box.

Shared edges in a net must have equal lengths. A four-by-two side attaches its four-centimeter edge to a four-centimeter edge of the base. A three-by-two side attaches its three-centimeter edge to a three-centimeter base edge. The two-centimeter dimensions become the box height after folding.

Find each face area before summing. Each four-by-three face has area twelve square centimeters, so that pair contributes twenty-four. Each four-by-two face has area eight, contributing sixteen. Each three-by-two face has area six, contributing twelve. The complete surface area is 24 + 16 + 12 = 52 square centimeters.

The compact expression is SA = 2lw + 2lh + 2wh. It summarizes the three pairs, but the net explains why those terms appear. Substituting four, three and two gives 2(4 × 3) + 2(4 × 2) + 2(3 × 2) = 52. Multiplying 4 × 3 × 2 instead gives twenty-four cubic centimeters of volume, a different measurement.

The net's outer perimeter is not the box's surface area. It measures boundary length in centimeters before folding, while surface area counts square centimeters covering faces. Fold lines are shared edges between faces; they do not add separate strips of area unless a real construction explicitly includes tabs or seams.

A box without a lid has only five of these outside rectangular faces. If the missing lid is a four-by-three face, subtract twelve from fifty-two to obtain forty square centimeters for the remaining five faces. State which face is absent. Removing a three-by-two face instead would subtract six and describe a different object.

Not every arrangement of six rectangles is a valid net. A face placed so that it folds onto an already occupied face creates overlap and leaves another side uncovered. Counting six pieces alone is insufficient. Trace their shared edges and actually fold a paper model or reason about each face's final position.

Surface-area problems need clear assumptions. A mathematical net represents zero-thickness faces and excludes extra material for glue tabs, overlap or trimming unless those are specified. For wrapping an actual box, those construction allowances may matter, but they are not silently included in the fifty-two-square-centimeter geometric surface area.

Fractional dimensions work in the same face-pair structure. A box with dimensions 1.5, 1 and 0.5 meters has paired face areas 1.5, 0.75 and 0.5 square meters. Doubling their sum gives 2(1.5 + 0.75 + 0.5) = 5.5 square meters. Its volume is only 0.75 cubic meter; units distinguish the measurements.

Worked missing-face correction: A learner sums only the three visible faces in a box sketch and obtains twenty-six square centimeters. A closed box also has the three opposite faces. The net displays all six, giving twice twenty-six, or fifty-two square centimeters.

Worked real-world open box (hypothetical): A cardboard tray with no lid measures 4 cm by 3 cm by 2 cm and stands on its four-by-three base. The base and four sides need 12 + 8 + 8 + 6 + 6 = 40 square centimeters of cardboard, not counting glue tabs. This agrees with removing one twelve-square-centimeter lid from the closed-box total.

Guided folding: Copy the net at a convenient scale, label every edge length and face area, and fold along shared edges. Identify the base, lid and four side faces before closing it. Check that each outside position is occupied once.

Independent performance: Draw a valid net for a different rectangular prism, calculate every face area and verify the total with the paired-face expression. Include fractional dimensions in a second example. Compare a closed and open version, state which face is removed and explain why volume, net perimeter and surface area answer different questions.