Stage 8 - Q1 - 2 marks
Find the volume of a cuboid 2 cm by 3 cm by 4 cm.
Hint
Multiply three dimensions.
Stage 8 - Q2 - 2 marks
A cuboid is 2 cm by 3 cm by 4 cm. List the areas of its three different face pairs.
Hint
Each pair of dimensions describes one type of face.
Stage 8 - Q3 - 2 marks
Find the surface area of a cuboid 2 cm by 3 cm by 4 cm.
Hint
Count both faces in every pair.
Stage 8 - Q4 - 3 marks
A 12 cm square sheet has area 144 cm². A proposed cuboid needs 150 cm² of faces. Can all faces fit without overlap? Explain.
Hint
Area gives a necessary first check.
Stage 8 - Q5 - 4 marks
A 3 cm by 4 cm by 5 cm cuboid has six separate faces. Find its surface area and volume, and state whether the area test alone proves it can be cut from a 12 cm square.
Hint
A positive area test still needs a layout.
Stage 8 - Q6 - 4 marks
Use the displayed three-row layout for a 3 cm by 6 cm by 6 cm cuboid. Verify that it fits a 12 cm square and find its volume.
Hint
Check the total width and height, not only the face areas.
Stage 8 - Q7 - 4 marks
A three-row packing places ab and ac faces side by side in each of two rows, then two bc faces side by side. Derive its required width and height.
Hint
The overall width is the wider row, while heights add.
Stage 8 - Q8 - 5 marks
Apply the three-row packing rule to a=2, b=5, c=7 cm. Does it fit a 12 cm square? Compare its volume with the 3 by 6 by 6 cuboid.
Hint
Check feasibility before comparing volumes.
Stage 8 - Q9 - 5 marks
A 4 by 4 by 7 cuboid has face area 144 cm² and volume 112 cm³. A pupil says that proves it can be made from a 12 cm square. Explain the missing evidence. Do not assume a layout exists.
Hint
Distinguish the amount of paper from the shape of available paper.
Stage 8 - Q10 - 6 marks
Restrict the three-row packing to a=3 cm and positive whole-number b,c. Prove that the largest volume within this family on a 12 cm square is 108 cm³. Exhibit a layout that attains it.
Hint
Use separate width and height inequalities to bound the product.