Nets of 3D Shapes

Worked example

1. 1

   $\text{end: pentagon, }n=5$ — Name the end first. A pentagon has 5 sides, so n = 5.

2. 2

   $5\text{ rectangles}$ — Each side of one end is joined to the other end by a rectangle, so there is one rectangle per side.

3. 3

   $5+2=7\text{ faces}$ — Add the 2 pentagon ends. This matches the rule n + 2.

Worked example

1. 1

   $\text{end: hexagon, }n=6$ — Name the end first. A hexagon has 6 sides, so n = 6.

2. 2

   $\text{edges round the ends}=2\times 6=12$ — Each hexagon end has 6 edges round it, and there are 2 ends.

3. 3

   $\text{edges joining the ends}=6$ — One edge runs from each corner of one end to the matching corner of the other end. 6 corners, so 6 edges.

4. 4

   $\text{edges}=12+6=18$ — Add the two groups. This matches the rule 3n = 3 × 6.

5. 5

   $\text{vertices}=2\times 6=12$ — Every corner sits on an end. Each end has 6 corners, and there are 2 ends. This matches 2n.

Worked example

1. 1

   $l=7,w=3,h=2$ — A cuboid 7 cm by 3 cm by 2 cm. Read off the three different lengths.

2. 2

   $l+w+h=7+3+2=12$ — Add one edge of each length: 12 cm.

3. 3

   $4\times 12=48\text{ cm}$ — Each length appears 4 times, so multiply by 4. If you had 60 cm of wire, 60 − 48 = 12 cm would be left.

Worked example

1. 1

   $4\times 3,4\times 2,3\times 2$ — List the three face sizes. A cuboid has two faces of each size, so you need 6 rectangles.

2. 2

   $\text{given: }4\times 3$ — Say the drawn face is the 4 cm by 3 cm base. Every new face must touch an edge of the same length.

3. 3

   $\text{on both }4\text{ cm edges: }4\times 2$ — Fold up the front and the back. They share the 4 cm edge, and their other side is the height, 2 cm.

4. 4

   $\text{on both }3\text{ cm edges: }3\times 2$ — Fold up the two ends. They share the 3 cm edge, and they are also 2 cm high.

5. 5

   $\text{beyond the back: }4\times 3$ — The lid. It folds over from the top edge of the back face, so it is the same size as the base.

6. 6

   $2+2+2=6\text{ faces}$ — Check: two of each size, six in all, and no two fold onto the same side.

Worked example

1. 1

   $3\times 8,4\times 8,5\times 8$ — One rectangle for each side of the triangle. Each is as long as the prism, 8 cm.

2. 2

   $\text{strip: }3+4+5=12\text{ cm wide}$ — Draw the three rectangles side by side in one strip, in the order 3, 4, 5, so they can wrap round the triangle.

3. 3

   $\text{2 triangles on the }4\text{ cm edges}$ — Attach one triangle to each short end of the 4 cm rectangle. They go on opposite sides, because the two ends of a prism are at opposite ends.

4. 4

   $3\text{ cm}\perp 4\text{ cm}$ — Draw the 4 cm side along the rectangle, and the 3 cm side at a right angle, next to the 3 cm rectangle. Joining their ends gives the 5 cm side without measuring.

Worked example

1. 1

   $\text{base: }6\times 6\text{ square}$ — Draw the square first. Every triangle sits on one of its sides.

2. 2

   $4\text{ triangles, base }6\text{ cm}$ — Put one triangle on each side of the square, pointing outwards, with its 6 cm side on the square.

3. 3

   $\text{tip: }4\text{ cm out from the middle}$ — Mark the tip 4 cm straight out from the middle of each side, then join it to both ends of that side.

4. 4

   $\text{sloping side}=\sqrt{3^2+4^2}=5\text{ cm}$ — Check with Pythagoras: half of 6 is 3, so each sloping side is 5 cm. All four triangles match, so their tips meet when folded.

Worked example

1. 1

   $V=30,l=5,w=2$ — A cuboid has volume 30 cm³ and a 5 cm by 2 cm base. The height is missing.

2. 2

   $l\times w=5\times 2=10$ — Work out the base area first: 10 cm².

3. 3

   $h=30÷10=3$ — Divide the volume by the base area. The height is 3 cm.

4. 4

   $5\times 2\times 3=30$ — Check by multiplying back. It gives the volume, so now draw the net of a 5 by 2 by 3 cuboid.

Worked example

1. 1

   $\text{edge}\times \text{edge}=25$ — Every face of a cube is the same square, and a square's area is edge × edge.

2. 2

   $\text{edge}=\sqrt{25}=5\text{ cm}$ — So the edge is the square root of the area.

3. 3

   $6\text{ squares, }5\times 5$ — Draw six 5 cm squares, for example a row of four with one above and one below the second square.

Worked example

1. 1

   $\text{row: }1,2,3,4$ — Four squares in a row wrap round the cube like a belt.

2. 2

   $1\leftrightarrow 3,2\leftrightarrow 4$ — In a straight row, squares two apart fold to opposite faces, because the square between them becomes a side.

3. 3

   $T\leftrightarrow B$ — The squares above and below the row become the top and the bottom, so they are opposite.

4. 4

   $\text{dice: }7-2=5$ — On a dice, opposite faces add up to 7. If square 1 shows 2, square 3 must show 5.