Problem list（15 problems）
- Volume of the cuboid? "Base area × height" works for any prism — square, triangular, or circular
- Volume of the cylinder? Cylinders and prisms alike are "base area × height"
- Volume of the cone? Any pointed solid is one third of the prism with the same base and height — pyramids and cones alike
- What is the total surface area? For surface area, do not count six separate faces — see three pairs of identical faces
- What is the total surface area? A curved surface flattens out if you cut it open: a cylinder's side becomes a rectangle, a cone's becomes a sector
- What is the angle at the centre of the sector? The whole of a cone's net rests on one fact: arc = base circumference
- What do the three "?" faces add up to? Hunting for which faces are opposite means fresh work for every net
- How many cubes have 2 painted faces? For N×N×N: 3 faces is always 8, 2 faces is 12×(N−2), 1 face is 6×(N−2)², 0 faces is (N−2)³
- What is the volume of the bottom part? Double the lengths and areas quadruple, volumes go up eightfold
- Volume of the solid made by one full turn? For a solid of revolution, "the farthest point from the axis is the radius"
- What is the volume of the solid produced? A solid of revolution is just cylinders added step by step
- How deep was the water? Every tilting problem rests on one fact: the volume does not change
- Volume of the stone? However complicated the shape, sinking it in water measures its volume — the very thing Archimedes noticed in his bath
- How deep will the water be? Levelling problems are always "all the water ÷ total base area"
- What is the volume of the box? In cut-and-fold problems the loss happens at both ends
Other ideas
- Shearing — moving a shape without changing its area
- Ratios — answers without fixing a single number
- Add, then subtract — an overlap is a counting problem
- Circles and sectors — multiply by π once, at the very end
- Chasing angles — there are only four tools
- Decide how to count, then count
- Write the same thing two different ways
- Fold it, turn it, copy it across
- Back to why the formula works