Nets of Other Solids

Opening a box produces rectangles, but other solids produce different flat pieces. A triangular prism needs triangles as well as rectangles. A cylinder contains a curved surface that unrolls smoothly. A cone opens into a shape resembling a slice of a circular sheet.

The same test applies throughout: the pieces must have the right shape, size, and connections to close the intended surface. We will connect each flat piece to the part of the solid it becomes.

Tetrahedra and Pyramids

A regular tetrahedron has four congruent equilateral triangular faces. One net has a central equilateral triangle with one congruent triangle attached to each side. Hold the central one as the base and fold the other three upwards until their free vertices meet.

Another tetrahedron net places the four triangles in a zigzag chain. These are the two distinct net forms for a regular tetrahedron when rotations and reflections are identified. In either form there are exactly four faces, even if glue tabs are added for the model.

A square pyramid has one square base and four triangular side faces. For a symmetric model, attach congruent isosceles triangles to the four base edges. Neighbouring triangular faces must have matching joining edges so that their free sides meet correctly.

Face shape alone is not enough. If the side triangles are too shallow, their tips may fail to meet above the centre as a genuine pyramid. For example, four equilateral triangles whose side matches the base edge do form a square pyramid, whereas arbitrarily chosen triangles need to be checked.

Triangular Prisms

For a right triangular prism, the two matching triangles are the end faces. Each side of a triangle corresponds to one rectangular side face. If the triangle sides are a, b, and c, and the prism length is L, the rectangles measure a by L, b by L, and c by L.

The three rectangles can form a strip. Attach one triangular end to a suitable edge on each opposite boundary of that strip. Matching the triangle’s side lengths to the rectangle widths is what allows the strip to wrap around it.

Unrolling a Cylinder

Wrap a rectangular sheet around a circular tube. One pair of opposite edges meets along a straight vertical seam. The distance once around the tube is the width of the unrolled rectangle, and the cylinder’s height becomes the rectangle’s other dimension.

If the base radius is r, the circumference is 2πr. A closed cylinder therefore needs a rectangle of dimensions 2πr by h and two circles of radius r. The rectangle width matches the whole circular boundary, not the diameter 2r.

Unrolling a Cone

Cut a right circular cone along a straight line from its apex to its base rim. Open its curved surface without stretching it. The resulting piece is a sector of a circle: two radii and the arc between them.

The sector centre becomes the cone’s apex. Its radius is the slant height s, measured down the cone’s surface from apex to rim. This is different from vertical height h, measured straight down inside the cone. The sector arc becomes the whole base circumference, so its length is 2πr.

Moving a cone’s apex sideways produces an oblique cone. Distances from that apex to different rim points are generally unequal, so its opened surface does not have the same simple circular-sector description. The neat sector rule belongs to the right circular cone, whose apex lies directly above the base centre.

Octahedra, Dodecahedra, and the Sphere

Join two square pyramids base to base. The shared squares become internal and are not outside faces. The eight triangular side faces form an octahedron. A regular octahedron has eight equilateral faces, twelve edges, and six vertices. It has 11 distinct nets under the usual rotation-and-reflection convention.

A regular dodecahedron uses twelve regular pentagonal faces. The many possible cutting choices explain why it has far more nets: 43,380. You do not need to draw them all to understand the main point: one solid can have many different unfoldings.

A sphere behaves differently. You cannot flatten an exact spherical surface onto one plane without stretching, tearing, or overlapping. Thin strips and small panels can approximate a globe, but they do not provide a perfect ordinary flat net of the sphere. The smooth curve in every direction is what makes paper wrapping wrinkle.

You can make an octahedron from eight equal equilateral triangles using the net below. Six triangles form a zigzag strip; one extra triangle is attached above the strip and one below. Fold along shared sides so that four triangles meet at each vertex. The two square-pyramid halves help you imagine the result, but neither hidden joining square belongs in this eight-triangle net.

Copy the net with a convenient triangle side such as 4 cm. Keep the joining edges uncut and add any glue tabs outside the eight faces. Fold gently and check that all free boundary edges pair up, without leaving a ninth face or an opening. The two green triangles close portions of the surface that the six-triangle strip alone would leave open.

Compare Curved Surfaces

Use a paper cylinder, a paper cone, and a ball to investigate flattening. Keep the investigation about the surface shape.

1. Open the cylinder along one straight seam and identify the rectangle.
2. Open the cone from apex to rim and identify the sector.
3. Try gently fitting a flat sheet around the ball without stretching it.
4. Compare the cylinder’s and cone’s clean openings with the ball’s folds or gaps.

Cylinders and cones can be developed into flat pieces along suitable cuts. A sphere requires distortion or approximation. This is a geometric distinction, not simply a problem of choosing a larger sheet.