Nets that fold into solids
Draw and interpret nets, matching faces and lengths while checking for overlap.
Core and Extended.
Before you begin
- Recognise solid faces.
- Calculate rectangular and triangular areas.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is the area of a 3 cm by 4 cm rectangle?
What you will learn
- Identify and draw nets of cubes, cuboids, prisms and pyramids.
- Match shared edges when folding.
- Use face dimensions to calculate a surface area.
A net opens every face
A net is a connected flat arrangement of all the faces of a solid that can fold without overlap. A cube needs six equal squares, but six squares in a strip will not form a cube: folding cannot create six distinct enclosing faces. Imagine folding around shared edges, or cut out a paper copy to check a doubtful arrangement.
In the illustrated cube net, four squares form a row and the remaining squares attach above and below the second square. That square can become the base; its neighbours rise around it and the far end closes the top. Tabs used for gluing are not faces and are excluded from surface area. Draw straight edges with a ruler and mark matching lengths.
Match the net to its cross-section
A cuboid net has three pairs of equal rectangles. For dimensions 3, 4 and 5 cm, the face pairs are 3 by 4, 3 by 5 and 4 by 5. A right triangular prism net has two congruent triangles and three rectangles whose widths match the three triangle sides; their other dimension is the prism's length.
A square-based pyramid net has one square with four triangles, each triangle's base matching a square side. The triangle's perpendicular face height is a sloping measurement on the solid, not necessarily the pyramid's vertical height. To find surface area from a net, add the areas of every face once. The net reveals boundaries; calculating volumes is developed in Mensuration.
Pause and explain
A square-based pyramid net has which faces?
Worked example
A closed cuboid has sides 3 cm, 4 cm and 5 cm. Describe its net and calculate the total area of its faces.
Show the worked solution
- Use two 3 × 4 rectangles, two 3 × 5 rectangles and two 4 × 5 rectangles.
- Their areas are 12, 15 and 20 cm² for one of each pair.
- Double the sum: 2(12 + 15 + 20) = 94 cm².
Answer Six matched rectangles; total face area 94 cm².
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Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Draw the illustrated cube net with each square side 2 cm. Find its face area.
Give me a hint
Draw six equal squares with matching fold edges.
Compare my reasoning
- Draw four 2 cm squares in a row, adding one above and one below the second.
- Each face has area 2 × 2 = 4 cm².
- All six faces give 6 × 4 = 24 cm²; exclude any glue tabs.
A valid cross-shaped net; 24 cm².
Look for these in your work
- Drew six equal faces with a ruler.
- Counted each face once.
Sketch a net for a right triangular prism with a 3–4–5 cm triangular cross-section and length 8 cm. Find its surface area.
Give me a hint
Use two right triangles and rectangles of widths 3, 4 and 5.
Compare my reasoning
- The two triangular faces total 2 × (3 × 4 ÷ 2) = 12 cm².
- The rectangular faces total 8(3 + 4 + 5) = 96 cm².
- Adding all five faces gives 108 cm²; match the triangle edges to rectangle widths.
Two 3–4–5 triangles and 3×8, 4×8, 5×8 rectangles; 108 cm².
Look for these in your work
- Matched all three cross-section edges.
- Included both end faces.
A pyramid has a 6 cm square base and four triangular faces, each with perpendicular face height 5 cm. How much card covers the closed solid, excluding tabs?
Give me a hint
Use the face height given, not a vertical solid height.
Compare my reasoning
- The square base area is 6 × 6 = 36 cm².
- Each triangle area is 6 × 5 ÷ 2 = 15 cm²; four total 60 cm².
- The closed net needs 36 + 60 = 96 cm².
96 cm².
Look for these in your work
- Used perpendicular height for each triangular face.
- Included the base and all four sides.
Common mistakes
- Accepting a net from face count alone.
- Leaving out a closed solid's base or end face.
- Confusing a triangle's face height with the pyramid's vertical height.
What can you explain now?
Is every connected arrangement of six equal squares a cube net?
Compare with the explanation
No.
The faces must fold into six distinct enclosing positions without overlap; being connected and having six squares is insufficient.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, draw a cuboid net and label its three pairs of equal rectangles. Fold a paper copy to check your arrangement.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.