Problem list
- a + b + c = ? Count the same area two ways — split into three, and all at once
- What is the side of the square? If the legs are a and b, the inscribed square's side is always a×b÷(a+b)
- How long is MN? "Join the midpoints of two sides and you get half the third side
- How long is the rope? Three logs or four, the curved part of a rope going once round is always one full circle
- How long is the pole's shadow? The sun casts parallel rays; a lamp spreads from a point
- How deep was the water? Every tilting problem rests on one fact: the volume does not change
- How far does the centre travel? However many sides the polygon has, the corner arcs always make one full circle
- How long is the line from B down to AC? Write the same area two ways — with two different base-and-height pairs — and set them equal
- How deep will the water be? Levelling problems are always "all the water ÷ total base area"
- How far from the left wall did it bounce? For a bounce, reflect and straighten
- How far is O, where the diagonals cross, from AD? The "hourglass" made by crossing diagonals is always similar, as long as there are parallel sides
- Where does the halving line cross the bottom edge? A rectangle is halved by any line through its centre
- If the rope is moved 1 m outwards, how much longer must it be? The famous result that the answer does not depend on the original radius
- What is the perimeter of the hexagon? 3
- How long is DC? "Same height, so area ratio = base ratio" works in both directions — lengths to areas, and areas back to lengths
- How long is the whole spiral? Multiply by π once, at the very end
- Perimeter of this shape? However many steps there are, the answer is the same
- How tall is the tree? Getting a height you cannot measure from a shadow you can
- How long is DE? The trick with ratios is to convert to "the whole of AB", not DB
- Perimeter of the sector? Forgetting to add the two radii is by far the most common mistake in perimeter problems
Area Problems Angle Problems Volume Problems Counting Problems Hard Problems Standard Problems