Nets of solids

A net is a flat pattern made of polygons that can be folded along its edges to form a 3D solid. Every box you ever opened was once a net before someone glued it. Nets are how packages get printed, shipped, and assembled.

Concept

Cube nets. A cube has $6$ square faces. To make a net, you flatten the cube so all $6$ squares lie on a plane, sharing edges that used to be folded.

There are exactly $11$ distinct nets of a cube (not counting rotations and reflections as different). The most common one looks like a cross or plus sign: a row of four squares with one square above and one below the second square. Other layouts shift the upper/lower squares to other positions.

Cuboid nets. A cuboid has $6$ rectangular faces, in $3$ matching pairs. The net is similar to the cube's but rectangles instead of squares.

Pyramid nets. A pyramid with a square base has one square and four triangles. The net is a square with one triangle on each side.

Prism nets. A triangular prism has two triangles and three rectangles. The net is the two triangles, three rectangles, sharing edges so they fold into the prism.

Drawing a net , procedure.

1. Identify all the faces of the solid.
2. Pick one face as the "base" and lay it flat.
3. Imagine "unfolding" each adjacent face along the edge it shares with the base.
4. Continue with each face until all are flat. Make sure no two faces overlap in the net.

Checking a net. Cut it out and try to fold. If it makes the solid without tearing or overlapping, the net is valid.

Surface area from a net. Since all faces lie flat, the total surface area of the solid equals the area of the net.

Example: surface area of a cube with edge $5$ cm is $6 \cdot 5^2 = 150$ cm$^2$.

Worked examples

Example 1. Sketch a net of a cube with edge $3$ cm (cross layout).

• Six squares of $3 \times 3 = 9$ cm$^2$ each, arranged in a cross.
• Total surface area: $54$ cm$^2$.

Example 2. Surface area of a cuboid $4 \times 5 \times 6$.

• Faces: $2(4 \cdot 5) + 2(5 \cdot 6) + 2(4 \cdot 6) = 40 + 60 + 48 = 148$ cm$^2$.

Example 3. Net of a square pyramid with base $4$ cm and slant height $5$ cm.

• One $4 \times 4$ square (area $16$).
• Four congruent isoceles triangles, each base $4$, slant height $5$, area $\frac{1}{2}(4)(5) = 10$.
• Total surface area: $16 + 4 \cdot 10 = 56$ cm$^2$.

Example 4. A triangular prism has equilateral triangular ends of side $4$ cm and length $10$ cm. Surface area?

• Two triangles, side $4$: area each $= \frac{\sqrt{3}}{4} \cdot 16 = 4\sqrt{3} \approx 6.93$ cm$^2$.
• Three rectangles $4 \times 10 = 40$ cm$^2$ each.
• Total: $2 \cdot 6.93 + 3 \cdot 40 = 13.86 + 120 \approx 133.86$ cm$^2$.

Try it yourself

1. Sketch all $11$ nets of a cube. (Hint: search for "$11$ cube nets" image to compare.)
2. Find the surface area of a cube with edge $8$ cm.
3. Find the surface area of a cuboid $2 \times 3 \times 5$.
4. Sketch a net of a triangular prism with sides $3, 4, 5$ (so it's a right triangle prism) and length $6$.
5. Find the surface area of a cylindrical can with radius $3$ cm and height $7$ cm. (Hint: net is rectangle + 2 circles.) Use $\pi \approx 3.14$.
6. A box has dimensions $10 \times 10 \times 30$. Find its surface area.
7. Sketch the net of a regular tetrahedron.
8. Net of a square pyramid: base $6$, slant height $8$. Surface area?

Activity

Empty the box. Carefully cut open an empty cereal box along the edges so it lies flat. You have a real, life-sized net. Measure each face's area, sum them , that is the surface area of the box. Now refold it. You have just lived through the meaning of a net.